# Finite posets¶

This module implements finite partially ordered sets. It defines:

 FinitePoset A class for finite posets FinitePosets_n A class for finite posets up to isomorphism (i.e. unlabeled posets) Poset() Construct a finite poset from various forms of input data. is_poset() Return True if a directed graph is acyclic and transitively reduced.

## List of Poset methods¶

Comparing, intervals and relations

 is_less_than() Return True if $$x$$ is strictly less than $$y$$ in the poset. is_greater_than() Return True if $$x$$ is strictly greater than $$y$$ in the poset. is_lequal() Return True if $$x$$ is less than or equal to $$y$$ in the poset. is_gequal() Return True if $$x$$ is greater than or equal to $$y$$ in the poset. compare_elements() Compare two element of the poset. closed_interval() Return the list of elements in a closed interval of the poset. open_interval() Return the list of elements in an open interval of the poset. relations() Return the list of relations in the poset. relations_iterator() Return an iterator over relations in the poset. order_filter() Return the upper set generated by elements. order_ideal() Return the lower set generated by elements.

Covering

 covers() Return True if y covers x. lower_covers() Return elements covered by given element. upper_covers() Return elements covering given element. cover_relations() Return the list of cover relations. lower_covers_iterator() Return an iterator over elements covered by given element. upper_covers_iterator() Return an iterator over elements covering given element. cover_relations_iterator() Return an iterator over cover relations of the poset.

Properties of the poset

 cardinality() Return the number of elements in the poset. height() Return the number of elements in a longest chain of the poset. width() Return the number of elements in a longest antichain of the poset. relations_number() Return the number of relations in the poset. dimension() Return the dimension of the poset. jump_number() Return the jump number of the poset. has_bottom() Return True if the poset has a unique minimal element. has_top() Return True if the poset has a unique maximal element. is_bounded() Return True if the poset has both unique minimal and unique maximal element. is_chain() Return True if the poset is totally ordered. is_connected() Return True if the poset is connected. is_graded() Return True if all maximal chains of the poset has same length. is_ranked() Return True if the poset has a rank function. is_rank_symmetric() Return True if the poset is rank symmetric. is_series_parallel() Return True if the poset can be built by ordinal sums and disjoint unions. is_greedy() Return True if all greedy linear extensions have equal number of jumps. is_jump_critical() Return True if removal of any element reduces the jump number. is_eulerian() Return True if the poset is Eulerian. is_incomparable_chain_free() Return True if the poset is (m+n)-free. is_slender() Return True if the poset is slender. is_sperner() Return True if the poset is Sperner. is_join_semilattice() Return True is the poset has a join operation. is_meet_semilattice() Return True if the poset has a meet operation.

Minimal and maximal elements

 bottom() Return the bottom element of the poset, if it exists. top() Return the top element of the poset, if it exists. maximal_elements() Return the list of the maximal elements of the poset. minimal_elements() Return the list of the minimal elements of the poset.

New posets from old ones

 disjoint_union() Return the disjoint union of the poset with other poset. ordinal_sum() Return the ordinal sum of the poset with other poset. product() Return the Cartesian product of the poset with other poset. ordinal_product() Return the ordinal product of the poset with other poset. rees_product() Return the Rees product of the poset with other poset. lexicographic_sum() Return the lexicographic sum of posets. star_product() Return the star product of the poset with other poset. with_bounds() Return the poset with bottom and top element adjoined. without_bounds() Return the poset with bottom and top element removed. dual() Return the dual of the poset. completion_by_cuts() Return the Dedekind-MacNeille completion of the poset. intervals_poset() Return the poset of intervals of the poset. connected_components() Return the connected components of the poset as subposets. factor() Return the decomposition of the poset as a Cartesian product. ordinal_summands() Return the ordinal summands of the poset. subposet() Return the subposet containing elements with partial order induced by this poset. random_subposet() Return a random subposet that contains each element with given probability. relabel() Return a copy of this poset with its elements relabelled. canonical_label() Return copy of the poset canonically (re)labelled to integers.

Chains & antichains

 is_chain_of_poset() Return True if elements in the given list are comparable. is_antichain_of_poset() Return True if elements in the given list are incomparable. chains() Return the chains of the poset. antichains() Return the antichains of the poset. maximal_chains() Return the maximal chains of the poset. maximal_antichains() Return the maximal antichains of the poset. antichains_iterator() Return an iterator over the antichains of the poset. random_maximal_chain() Return a random maximal chain. random_maximal_antichain() Return a random maximal antichain.

Drawing

 show() Display the Hasse diagram of the poset. plot() Return a Graphic object corresponding the Hasse diagram of the poset. graphviz_string() Return a representation in the DOT language, ready to render in graphviz.

Comparing posets

 is_isomorphic() Return True if both posets are isomorphic. is_induced_subposet() Return True if given poset is an induced subposet of this poset.

Polynomials

 chain_polynomial() Return the chain polynomial of the poset. characteristic_polynomial() Return the characteristic polynomial of the poset. f_polynomial() Return the f-polynomial of the poset. flag_f_polynomial() Return the flag f-polynomial of the poset. h_polynomial() Return the h-polynomial of the poset. flag_h_polynomial() Return the flag h-polynomial of the poset. order_polynomial() Return the order polynomial of the poset. zeta_polynomial() Return the zeta polynomial of the poset. kazhdan_lusztig_polynomial() Return the Kazhdan-Lusztig polynomial of the poset. coxeter_polynomial() Return the characteristic polynomial of the Coxeter transformation. degree_polynomial() Return the generating polynomial of degrees of vertices in the Hasse diagram. p_partition_enumerator() Return a $$P$$-partition enumerator of the poset.

Polytopes

 chain_polytope() Return the chain polytope of the poset. order_polytope() Return the order polytope of the poset.

Graphs

 hasse_diagram() Return the Hasse diagram of the poset as a directed graph. cover_relations_graph() Return the (undirected) graph of cover relations. comparability_graph() Return the comparability graph of the poset. incomparability_graph() Return the incomparability graph of the poset. frank_network() Return Frank’s network of the poset. linear_extensions_graph() Return the linear extensions graph of the poset.

Linear extensions

 is_linear_extension() Return True if the given list is a linear extension of the poset. linear_extension() Return a linear extension of the poset. linear_extensions() Return the enumerated set of all the linear extensions of the poset. promotion() Return the (extended) promotion on the linear extension of the poset. evacuation() Return evacuation on the linear extension associated to the poset. with_linear_extension() Return a copy of self with a different default linear extension. random_linear_extension() Return a random linear extension.

Matrices

 lequal_matrix() Computes the matrix whose (i,j) entry is 1 if self.linear_extension()[i] < self.linear_extension()[j] and 0 otherwise. moebius_function() Return the value of Möbius function of given elements in the poset. moebius_function_matrix() Return a matrix whose (i,j) entry is the value of the Möbius function evaluated at self.linear_extension()[i] and self.linear_extension()[j]. coxeter_transformation() Return the matrix of the Auslander-Reiten translation acting on the Grothendieck group of the derived category of modules. coxeter_smith_form() Return the Smith form of the Coxeter transformation.

Miscellanous

 sorted() Return given list sorted by the poset. isomorphic_subposets() Return all subposets isomorphic to another poset. isomorphic_subposets_iterator() Return an iterator over the subposets isomorphic to another poset. has_isomorphic_subposet() Return True if the poset contains a subposet isomorphic to another poset. list() List the elements of the poset. cuts() Return the cuts of the given poset. dilworth_decomposition() Return a partition of the points into the minimal number of chains. greene_shape() Computes the Greene-Kleitman partition aka Greene shape of the poset self. incidence_algebra() Return the incidence algebra of self. is_EL_labelling() Return whether f is an EL labelling of the poset. isomorphic_subposets_iterator() Return an iterator over the subposets isomorphic to another poset. isomorphic_subposets() Return all subposets isomorphic to another poset. level_sets() Return elements grouped by maximal number of cover relations from a minimal element. order_complex() Return the order complex associated to this poset. random_order_ideal() Return a random order ideal of self with uniform probability. rank() Return the rank of an element, or the rank of the poset. rank_function() Return a rank function of the poset, if it exists. unwrap() Unwraps an element of this poset.

## Classes and functions¶

class sage.combinat.posets.posets.FinitePoset(hasse_diagram, elements, category, facade, key)

A (finite) $$n$$-element poset constructed from a directed acyclic graph.

INPUT:

• hasse_diagram – an instance of FinitePoset, or a DiGraph that is transitively-reduced, acyclic, loop-free, and multiedge-free.
• elements – an optional list of elements, with element[i] corresponding to vertex i. If elements is None, then it is set to be the vertex set of the digraph. Note that if this option is set, then elements is considered as a specified linear extension of the poset and the $$linear_extension$$ attribute is set.
• categoryFinitePosets, or a subcategory thereof.
• facade – a boolean or None (default); whether the FinitePoset’s elements should be wrapped to make them aware of the Poset they belong to.
• If facade = True, the FinitePoset’s elements are exactly those given as input.
• If facade = False, the FinitePoset’s elements will become PosetElement objects.
• If facade = None (default) the expected behaviour is the behaviour of facade = True, unless the opposite can be deduced from the context (i.e. for instance if a FinitePoset is built from another FinitePoset, itself built with facade = False)
• key – any hashable value (default: None).

EXAMPLES:

sage: uc = [[2,3], [], , , , [3,4]]
sage: from sage.combinat.posets.posets import FinitePoset
sage: P = FinitePoset(DiGraph(dict([[i,uc[i]] for i in range(len(uc))])), facade=False); P
Finite poset containing 6 elements
sage: P.cover_relations()
[[5, 4], [5, 3], [4, 1], [0, 2], [0, 3], [2, 1], [3, 1]]
sage: TestSuite(P).run()
sage: P.category()
Category of finite enumerated posets
sage: P.__class__
<class 'sage.combinat.posets.posets.FinitePoset_with_category'>

sage: Q = sage.combinat.posets.posets.FinitePoset(P, facade = False); Q
Finite poset containing 6 elements

sage: Q is P
True


We keep the same underlying Hasse diagram, but change the elements:

sage: Q = sage.combinat.posets.posets.FinitePoset(P, elements=[1,2,3,4,5,6], facade=False); Q
Finite poset containing 6 elements with distinguished linear extension
sage: Q.cover_relations()
[[1, 2], [1, 5], [2, 6], [3, 4], [3, 5], [4, 6], [5, 6]]


sage: P = Poset(DiGraph({'a':['b'],'b':['c'],'c':['d']}), facade=False)
sage: P.category()
Category of finite enumerated posets
sage: parent(P) is P
True

sage: Q.category()
Category of facade finite enumerated posets
sage: parent(Q) is str
True
sage: TestSuite(Q).run(skip = ['_test_an_element']) # is_parent_of is not yet implemented


sage: PQ = Poset(P, facade=True)
sage: PQ.category()
Category of facade finite enumerated posets
sage: parent(PQ) is str
True
sage: PQ is Q
True


sage: QP = Poset(Q, facade = False)
sage: QP.category()
Category of finite enumerated posets
sage: parent(QP) is QP
True


Note

A class that inherits from this class needs to define Element. This is the class of the elements that the inheriting class contains. For example, for this class, FinitePoset, Element is PosetElement. It can also define _dual_class which is the class of dual posets of this class. E.g. FiniteMeetSemilattice._dual_class is FiniteJoinSemilattice.

Element
antichains(element_constructor=<class 'list'>)

Return the antichains of the poset.

An antichain of a poset is a set of elements of the poset that are pairwise incomparable.

INPUT:

• element_constructor – a function taking an iterable as argument (default: list)

OUTPUT:

The enumerated set (of type PairwiseCompatibleSubsets) of all antichains of the poset, each of which is given as an element_constructor.

EXAMPLES:

sage: A = posets.PentagonPoset().antichains(); A
Set of antichains of Finite lattice containing 5 elements
sage: list(A)
[[], , , [1, 2], [1, 3], , , ]
sage: A.cardinality()
8
sage: A
[1, 2]


To get the antichains as, say, sets, one may use the element_constructor option:

sage: list(posets.ChainPoset(3).antichains(element_constructor=set))
[set(), {0}, {1}, {2}]


To get the antichains of a given size one can currently use:

sage: list(A.elements_of_depth_iterator(2))
[[1, 2], [1, 3]]


Eventually the following syntax will be accepted:

sage: A.subset(size = 2) # todo: not implemented


Note

Internally, this uses sage.combinat.subsets_pairwise.PairwiseCompatibleSubsets and SearchForest. At this point, iterating through this set is about twice slower than using antichains_iterator() (tested on posets.AntichainPoset(15)). The algorithm is the same (depth first search through the tree), but antichains_iterator() manually inlines things which apparently avoids some infrastructure overhead.

On the other hand, this returns a full featured enumerated set, with containment testing, etc.

antichains_iterator()

Return an iterator over the antichains of the poset.

EXAMPLES:

sage: it = posets.PentagonPoset().antichains_iterator(); it
<generator object ...antichains_iterator at ...>
sage: next(it), next(it)
([], )

bottom()

Return the unique minimal element of the poset, if it exists.

EXAMPLES:

sage: P = Poset({0:,1:,2:,3:,4:[]})
sage: P.bottom() is None
True
sage: Q = Poset({0:,1:[]})
sage: Q.bottom()
0


canonical_label(algorithm=None)

Return the unique poset on the labels $$\{0, \ldots, n-1\}$$ (where $$n$$ is the number of elements in the poset) that is isomorphic to this poset and invariant in the isomorphism class.

INPUT:

• algorithm – string (optional); a parameter forwarded to underlying graph function to select the algorithm to use

EXAMPLES:

sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
sage: P.list()
[1, 2, 3, 4, 6, 12]
sage: Q = P.canonical_label()
sage: sorted(Q.list())
[0, 1, 2, 3, 4, 5]
sage: Q.is_isomorphic(P)
True


Canonical labeling of (semi)lattice returns (semi)lattice:

sage: D = DiGraph({'a':['b','c']})
sage: P = Poset(D)
sage: ML = MeetSemilattice(D)
sage: P.canonical_label()
Finite poset containing 3 elements
sage: ML.canonical_label()
Finite meet-semilattice containing 3 elements


cardinality()

Return the number of elements in the poset.

EXAMPLES:

sage: Poset([[1,2,3],,,,[]]).cardinality()
5


degree_polynomial() for a more refined invariant

chain_polynomial()

Return the chain polynomial of the poset.

The coefficient of $$q^k$$ is the number of chains of $$k$$ elements in the poset. List of coefficients of this polynomial is also called a f-vector of the poset.

Note

This is not what has been called the chain polynomial in [St1986]. The latter is identical with the order polynomial in SageMath (order_polynomial()).

EXAMPLES:

sage: P = posets.ChainPoset(3)
sage: t = P.chain_polynomial(); t
q^3 + 3*q^2 + 3*q + 1
sage: t(1) == len(list(P.chains()))
True

sage: P = posets.BooleanLattice(3)
sage: P.chain_polynomial()
6*q^4 + 18*q^3 + 19*q^2 + 8*q + 1

sage: P = posets.AntichainPoset(5)
sage: P.chain_polynomial()
5*q + 1

chain_polytope()

Return the chain polytope of the poset self.

The chain polytope of a finite poset $$P$$ is defined as the subset of $$\RR^P$$ consisting of all maps $$x : P \to \RR$$ satisfying

$x(p) \geq 0 \mbox{ for all } p \in P,$

and

$x(p_1) + x(p_2) + \ldots + x(p_k) \leq 1 \mbox{ for all chains } p_1 < p_2 < \ldots < p_k \mbox{ in } P.$

This polytope was defined and studied in [St1986].

EXAMPLES:

sage: P = posets.AntichainPoset(3)
sage: Q = P.chain_polytope();Q
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 8 vertices
sage: P = posets.PentagonPoset()
sage: Q = P.chain_polytope();Q
A 5-dimensional polyhedron in ZZ^5 defined as the convex hull of 8 vertices

chains(element_constructor=<class 'list'>, exclude=None)

Return the chains of the poset.

A chain of a poset is a set of elements of the poset that are pairwise comparable.

INPUT:

• element_constructor – a function taking an iterable as argument (default: list)
• exclude – elements of the poset to be excluded (default: None)

OUTPUT:

The enumerated set (of type PairwiseCompatibleSubsets) of all chains of the poset, each of which is given as an element_constructor.

EXAMPLES:

sage: C = posets.PentagonPoset().chains(); C
Set of chains of Finite lattice containing 5 elements
sage: list(C)
[[], , [0, 1], [0, 1, 4], [0, 2], [0, 2, 3], [0, 2, 3, 4], [0, 2, 4], [0, 3], [0, 3, 4], [0, 4], , [1, 4], , [2, 3], [2, 3, 4], [2, 4], , [3, 4], ]


Exclusion of elements, tuple (instead of list) as constructor:

sage: P = Poset({1: [2, 3], 2: , 3: [4, 5]})
sage: list(P.chains(element_constructor=tuple, exclude=))
[(), (1,), (1, 2), (1, 2, 4), (1, 4), (1, 5), (2,), (2, 4), (4,), (5,)]


To get the chains of a given size one can currently use:

sage: list(C.elements_of_depth_iterator(2))
[[0, 1], [0, 2], [0, 3], [0, 4], [1, 4], [2, 3], [2, 4], [3, 4]]


Eventually the following syntax will be accepted:

sage: C.subset(size = 2) # todo: not implemented

characteristic_polynomial()

Return the characteristic polynomial of the poset.

The poset is expected to be graded and have a bottom element.

If $$P$$ is a graded poset with rank $$n$$ and a unique minimal element $$\hat{0}$$, then the characteristic polynomial of $$P$$ is defined to be

$\sum_{x \in P} \mu(\hat{0}, x) q^{n-\rho(x)} \in \ZZ[q],$

where $$\rho$$ is the rank function, and $$\mu$$ is the Möbius function of $$P$$.

See section 3.10 of [EnumComb1].

EXAMPLES:

sage: P = posets.DiamondPoset(5)
sage: P.characteristic_polynomial()
q^2 - 3*q + 2

sage: P = Poset({1: [2, 3], 2: , 3: , 4: , 5: , 6: })
sage: P.characteristic_polynomial()
q^4 - 2*q^3 + q

closed_interval(x, y)

Return the list of elements $$z$$ such that $$x \le z \le y$$ in the poset.

EXAMPLES:

sage: P = Poset((divisors(1000), attrcall("divides")))
sage: P.closed_interval(2, 100)
[2, 4, 10, 20, 50, 100]

comparability_graph()

Return the comparability graph of the poset.

The comparability graph is an undirected graph where vertices are the elements of the poset and there is an edge between two vertices if they are comparable in the poset.

EXAMPLES:

sage: Y = Poset({1: , 2: [3, 4]})
sage: g = Y.comparability_graph(); g
Comparability graph on 4 vertices
sage: Y.compare_elements(1, 3) is not None
True
sage: g.has_edge(1, 3)
True

compare_elements(x, y)

Compare $$x$$ and $$y$$ in the poset.

• If $$x < y$$, return -1.
• If $$x = y$$, return 0.
• If $$x > y$$, return 1.
• If $$x$$ and $$y$$ are not comparable, return None.

EXAMPLES:

sage: P = Poset([[1, 2], , , , []])
sage: P.compare_elements(0, 0)
0
sage: P.compare_elements(0, 4)
-1
sage: P.compare_elements(4, 0)
1
sage: P.compare_elements(1, 2) is None
True

completion_by_cuts()

Return the completion by cuts of self.

This is the smallest lattice containing the poset. This is also called the Dedekind-MacNeille completion.

OUTPUT:

• a finite lattice

EXAMPLES:

sage: P = posets.PentagonPoset()
sage: P.completion_by_cuts().is_isomorphic(P)
True

sage: Y = Poset({1: , 2: [3, 4]})
sage: trafficsign = LatticePoset({1: , 2: [3, 4], 3: , 4: })
sage: L = Y.completion_by_cuts()
sage: L.is_isomorphic(trafficsign)
True

sage: P = posets.SymmetricGroupBruhatOrderPoset(3)
sage: Q = P.completion_by_cuts(); Q
Finite lattice containing 7 elements

connected_components()

Return the connected components of the poset as subposets.

EXAMPLES:

sage: P = Poset({1: [2, 3], 3: [4, 5], 6: [7, 8]})
sage: parts = sorted(P.connected_components(), key=len); parts
[Finite poset containing 3 elements,
Finite poset containing 5 elements]
sage: parts.cover_relations()
[[6, 7], [6, 8]]

cover_relations()

Return the list of pairs [x, y] of elements of the poset such that y covers x.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:, 4:[]})
sage: P.cover_relations()
[[1, 2], [0, 2], [2, 3], [3, 4]]

cover_relations_graph()

Return the (undirected) graph of cover relations.

EXAMPLES:

sage: P = Poset({0: [1, 2], 1: , 2: })
sage: G = P.cover_relations_graph(); G
Graph on 4 vertices
sage: G.has_edge(3, 1), G.has_edge(3, 0)
(True, False)

cover_relations_iterator()

Return an iterator over the cover relations of the poset.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:, 4:[]})
sage: type(P.cover_relations_iterator())
<... 'generator'>
sage: [z for z in P.cover_relations_iterator()]
[[1, 2], [0, 2], [2, 3], [3, 4]]

covers(x, y)

Return True if y covers x and False otherwise.

Element $$y$$ covers $$x$$ if $$x < y$$ and there is no $$z$$ such that $$x < z < y$$.

EXAMPLES:

sage: P = Poset([[1,5], [2,6], , , [], [6,3], ])
sage: P.covers(1, 6)
True
sage: P.covers(1, 4)
False
sage: P.covers(1, 5)
False

coxeter_polynomial()

Return the Coxeter polynomial of the poset.

OUTPUT:

a polynomial in one variable

The output is the characteristic polynomial of the Coxeter transformation. This polynomial only depends on the derived category of modules on the poset.

EXAMPLES:

sage: P = posets.PentagonPoset()
sage: P.coxeter_polynomial()
x^5 + x^4 + x + 1

sage: p = posets.SymmetricGroupWeakOrderPoset(3)
sage: p.coxeter_polynomial()
x^6 + x^5 - x^3 + x + 1

coxeter_smith_form(algorithm='singular')

Return the Smith normal form of $$x$$ minus the Coxeter transformation matrix.

INPUT:

• algorithm – optional (default 'singular'), possible values are 'singular', 'sage', 'gap', 'pari', 'maple', 'magma', 'fricas'

Beware that speed depends very much on the choice of algorithm. Sage is rather slow, Singular is faster and Pari is fast at least for small sizes.

OUTPUT:

• list of polynomials in one variable, each one dividing the next one

The output list is a refinement of the characteristic polynomial of the Coxeter transformation, which is its product. This list of polynomials only depends on the derived category of modules on the poset.

EXAMPLES:

sage: P = posets.PentagonPoset()
sage: P.coxeter_smith_form()
[1, 1, 1, 1, x^5 + x^4 + x + 1]

sage: P = posets.DiamondPoset(7)
sage: prod(P.coxeter_smith_form()) == P.coxeter_polynomial()
True


coxeter_transformation(), coxeter_matrix()

coxeter_transformation()

Return the Coxeter transformation of the poset.

OUTPUT:

a square matrix with integer coefficients

The output is the matrix of the Auslander-Reiten translation acting on the Grothendieck group of the derived category of modules on the poset, in the basis of simple modules. This matrix is usually called the Coxeter transformation.

EXAMPLES:

sage: posets.PentagonPoset().coxeter_transformation()
[ 0  0  0  0 -1]
[ 0  0  0  1 -1]
[ 0  1  0  0 -1]
[-1  1  1  0 -1]
[-1  1  0  1 -1]

cuts()

Return the list of cuts of the poset self.

A cut is a subset $$A$$ of self such that the set of lower bounds of the set of upper bounds of $$A$$ is exactly $$A$$.

The cuts are computed here using the maximal independent sets in the auxiliary graph defined as $$P \times [0,1]$$ with an edge from $$(x, 0)$$ to $$(y, 1)$$ if and only if $$x \not\geq_P y$$. See the end of section 4 in [JRJ94].

EXAMPLES:

sage: P = posets.AntichainPoset(3)
sage: Pc = P.cuts()
sage: [list(c) for c in Pc]
[, [], [0, 1, 2], , ]
sage: Pc
frozenset({0})

degree_polynomial()

Return the generating polynomial of degrees of vertices in self.

This is the sum

$\sum_{v \in P} x^{\operatorname{in}(v)} y^{\operatorname{out}(v)},$

where in(v) and out(v) are the number of incoming and outgoing edges at vertex $$v$$ in the Hasse diagram of $$P$$.

Because this polynomial is multiplicative for Cartesian product of posets, it is useful to help see if the poset can be isomorphic to a Cartesian product.

EXAMPLES:

sage: P = posets.PentagonPoset()
sage: P.degree_polynomial()
x^2 + 3*x*y + y^2

sage: P = posets.BooleanLattice(4)
sage: P.degree_polynomial().factor()
(x + y)^4


cardinality() for the value at $$(x, y) = (1, 1)$$

dilworth_decomposition()

Return a partition of the points into the minimal number of chains.

According to Dilworth’s theorem, the points of a poset can be partitioned into $$\alpha$$ chains, where $$\alpha$$ is the cardinality of its largest antichain. This method returns such a partition.

ALGORITHM:

We build a bipartite graph in which a vertex $$v$$ of the poset is represented by two vertices $$v^-,v^+$$. For any two $$u,v$$ such that $$u<v$$ in the poset we add an edge $$v^+u^-$$.

A matching in this graph is equivalent to a partition of the poset into chains: indeed, a chain $$v_1...v_k$$ gives rise to the matching $$v_1^+v_2^-,v_2^+v_3^-,...$$, and from a matching one can build the union of chains.

According to Dilworth’s theorem, the number of chains is equal to $$\alpha$$ (the posets’ width).

EXAMPLES:

sage: p = posets.BooleanLattice(4)
sage: p.width()
6
sage: p.dilworth_decomposition()  # random
[[7, 6, 4], [11, 3], [12, 8, 0], [13, 9, 1], [14, 10, 2], [15, 5]]


level_sets() to return elements grouped to antichains.

dimension(certificate=False)

Return the dimension of the Poset.

The (Dushnik-Miller) dimension of a poset is the minimal number of total orders so that the poset is their “intersection”. More precisely, the dimension of a poset defined on a set $$X$$ of points is the smallest integer $$n$$ such that there exist linear extensions $$P_1,...,P_n$$ of $$P$$ satisfying:

$u\leq_P v\ \text{if and only if }\ \forall i, u\leq_{P_i} v$

INPUT:

• certificate (boolean; default:False) – whether to return an integer (the dimension) or a certificate, i.e. a smallest set of linear extensions.

Note

The speed of this function greatly improves when more efficient MILP solvers (e.g. Gurobi, CPLEX) are installed. See MixedIntegerLinearProgram for more information.

Note

Prior to version 8.3 this returned only realizer with certificate=True. Now it returns a pair having a realizer as the second element. See trac ticket #25588 for details.

ALGORITHM:

As explained [FT00], the dimension of a poset is equal to the (weak) chromatic number of a hypergraph. More precisely:

Let $$inc(P)$$ be the set of (ordered) pairs of incomparable elements of $$P$$, i.e. all $$uv$$ and $$vu$$ such that $$u\not \leq_P v$$ and $$v\not \leq_P u$$. Any linear extension of $$P$$ is a total order on $$X$$ that can be seen as the union of relations from $$P$$ along with some relations from $$inc(P)$$. Thus, the dimension of $$P$$ is the smallest number of linear extensions of $$P$$ which cover all points of $$inc(P)$$.

Consequently, $$dim(P)$$ is equal to the chromatic number of the hypergraph $$\mathcal H_{inc}$$, where $$\mathcal H_{inc}$$ is the hypergraph defined on $$inc(P)$$ whose sets are all $$S\subseteq inc(P)$$ such that $$P\cup S$$ is not acyclic.

We solve this problem through a Mixed Integer Linear Program.

The problem is known to be NP-complete.

EXAMPLES:

We create a poset, compute a set of linear extensions and check that we get back the poset from them:

sage: P = Poset([[1,4], , [4,5,3], , [], , []])
sage: P.dimension()
3
sage: dim, L = P.dimension(certificate=True)
sage: L  # random -- architecture-dependent
[[0, 2, 4, 5, 1, 3, 6], [2, 5, 0, 1, 3, 4, 6], [0, 1, 2, 3, 5, 6, 4]]
sage: Poset( (L, lambda x, y: all(l.index(x) < l.index(y) for l in L)) ) == P
True


According to Schnyder’s theorem, the incidence poset (of height 2) of a graph has dimension $$\leq 3$$ if and only if the graph is planar:

sage: G = graphs.CompleteGraph(4)
sage: P = Poset(DiGraph({(u,v):[u,v] for u,v,_ in G.edges()}))
sage: P.dimension()
3

sage: G = graphs.CompleteBipartiteGraph(3,3)
sage: P = Poset(DiGraph({(u,v):[u,v] for u,v,_ in G.edges()}))
sage: P.dimension() # not tested - around 4s with CPLEX
4

disjoint_union(other, labels='pairs')

Return a poset isomorphic to disjoint union (also called direct sum) of the poset with other.

The disjoint union of $$P$$ and $$Q$$ is a poset that contains every element and relation from both $$P$$ and $$Q$$, and where every element of $$P$$ is incomparable to every element of $$Q$$.

Mathematically, it is only defined when $$P$$ and $$Q$$ have no common element; here we force that by giving them different names in the resulting poset.

INPUT:

• other, a poset.
• labels - (defaults to ‘pairs’) If set to ‘pairs’, each element v in this poset will be named (0,v) and each element u in other will be named (1,u) in the result. If set to ‘integers’, the elements of the result will be relabeled with consecutive integers.

EXAMPLES:

sage: P1 = Poset({'a': 'b'})
sage: P2 = Poset({'c': 'd'})
sage: P = P1.disjoint_union(P2); P
Finite poset containing 4 elements
sage: sorted(P.cover_relations())
[[(0, 'a'), (0, 'b')], [(1, 'c'), (1, 'd')]]
sage: P = P1.disjoint_union(P2, labels='integers')
sage: P.cover_relations()
[[2, 3], [0, 1]]

sage: N5 = posets.PentagonPoset(); N5
Finite lattice containing 5 elements
sage: N5.disjoint_union(N5)  # Union of lattices is not a lattice
Finite poset containing 10 elements


We show how to get literally direct sum with elements untouched:

sage: P = P1.disjoint_union(P2).relabel(lambda x: x)
sage: sorted(P.cover_relations())
[['a', 'b'], ['c', 'd']]

dual()

Return the dual poset of the given poset.

In the dual of a poset $$P$$ we have $$x \le y$$ iff $$y \le x$$ in $$P$$.

EXAMPLES:

sage: P = Poset({1: [2, 3], 3: })
sage: P.cover_relations()
[[1, 2], [1, 3], [3, 4]]
sage: Q = P.dual()
sage: Q.cover_relations()
[[4, 3], [3, 1], [2, 1]]


Dual of a lattice is a lattice; dual of a meet-semilattice is join-semilattice and vice versa. Also the dual of a (non-)facade poset is again (non-)facade:

sage: V = MeetSemilattice({1: [2, 3]}, facade=False)
sage: A = V.dual(); A
Finite join-semilattice containing 3 elements
sage: A(2) < A(1)
True

evacuation()

Compute evacuation on the linear extension associated to the poset self.

OUTPUT:

• an isomorphic poset, with the same default linear extension

Evacuation is defined on a poset self of size $$n$$ by applying the evacuation operator $$(\tau_1 \cdots \tau_{n-1}) (\tau_1 \cdots \tau_{n-2}) \cdots (\tau_1)$$, to the default linear extension $$\pi$$ of self (see evacuation()), and relabeling self accordingly. For more details see [Stan2009].

EXAMPLES:

sage: P = Poset(([1,2], [[1,2]]), linear_extension=True, facade=False)
sage: P.evacuation()
Finite poset containing 2 elements with distinguished linear extension
sage: P.evacuation() == P
True

sage: P = Poset(([1,2,3,4,5,6,7], [[1,2],[1,4],[2,3],[2,5],[3,6],[4,7],[5,6]]), linear_extension=True, facade=False)
sage: P.list()
[1, 2, 3, 4, 5, 6, 7]
sage: Q = P.evacuation(); Q
Finite poset containing 7 elements with distinguished linear extension
sage: Q.cover_relations()
[[1, 2], [1, 3], [2, 5], [3, 4], [3, 6], [4, 7], [6, 7]]


Note that the results depend on the linear extension associated to the poset:

sage: P = Poset(([1,2,3,4,5,6,7], [[1,2],[1,4],[2,3],[2,5],[3,6],[4,7],[5,6]]))
sage: P.list()
[1, 2, 3, 5, 6, 4, 7]
sage: Q = P.evacuation(); Q
Finite poset containing 7 elements with distinguished linear extension
sage: Q.cover_relations()
[[1, 2], [1, 5], [2, 3], [5, 6], [5, 4], [6, 7], [4, 7]]


Here is an example of a poset where the elements are not labelled by $$\{1,2,\ldots,n\}$$:

sage: P = Poset((divisors(15), attrcall("divides")), linear_extension = True)
sage: P.list()
[1, 3, 5, 15]
sage: Q = P.evacuation(); Q
Finite poset containing 4 elements with distinguished linear extension
sage: Q.cover_relations()
[[1, 3], [1, 5], [3, 15], [5, 15]]


AUTHOR:

• Anne Schilling (2012-02-18)
f_polynomial()

Return the $$f$$-polynomial of the poset.

The poset is expected to be bounded.

This is the $$f$$-polynomial of the order complex of the poset minus its bounds.

The coefficient of $$q^i$$ is the number of chains of $$i+1$$ elements containing both bounds of the poset.

Note

This is slightly different from the fPolynomial method in Macaulay2.

EXAMPLES:

sage: P = posets.DiamondPoset(5)
sage: P.f_polynomial()
3*q^2 + q

sage: P = Poset({1: [2, 3], 2: , 3: , 4: , 5: , 6: })
sage: P.f_polynomial()
q^4 + 4*q^3 + 5*q^2 + q

factor()

Factor the poset as a Cartesian product of smaller posets.

This only works for connected posets for the moment.

The decomposition of a connected poset as a Cartesian product of posets (prime in the sense that they cannot be written as Cartesian products) is unique up to reordering and isomorphism.

OUTPUT:

a list of posets

EXAMPLES:

sage: P = posets.PentagonPoset()
sage: Q = P*P
sage: Q.factor()
[Finite poset containing 5 elements,
Finite poset containing 5 elements]

sage: P1 = posets.ChainPoset(3)
sage: P2 = posets.ChainPoset(7)
sage: P1.factor()
[Finite lattice containing 3 elements]
sage: (P1 * P2).factor()
[Finite poset containing 7 elements,
Finite poset containing 3 elements]

sage: P = posets.TamariLattice(4)
sage: (P*P).factor()
[Finite poset containing 14 elements,
Finite poset containing 14 elements]


REFERENCES:

 [Feig1986] Joan Feigenbaum, Directed Cartesian-Product Graphs have Unique Factorizations that can be computed in Polynomial Time, Discrete Applied Mathematics 15 (1986) 105-110 doi:10.1016/0166-218X(86)90023-5
flag_f_polynomial()

Return the flag $$f$$-polynomial of the poset.

The poset is expected to be bounded and ranked.

This is the sum, over all chains containing both bounds, of a monomial encoding the ranks of the elements of the chain.

More precisely, if $$P$$ is a bounded ranked poset, then the flag $$f$$-polynomial of $$P$$ is defined as the polynomial

$\begin{split}\sum_{\substack{p_0 < p_1 < \ldots < p_k, \\ p_0 = \min P, \ p_k = \max P}} x_{\rho(p_1)} x_{\rho(p_2)} \cdots x_{\rho(p_k)} \in \ZZ[x_1, x_2, \cdots, x_n]\end{split}$

where $$\min P$$ and $$\max P$$ are (respectively) the minimum and the maximum of $$P$$, where $$\rho$$ is the rank function of $$P$$ (normalized to satisfy $$\rho(\min P) = 0$$), and where $$n$$ is the rank of $$\max P$$. (Note that the indeterminate $$x_0$$ doesn’t actually appear in the polynomial.)

For technical reasons, the polynomial is returned in the slightly larger ring $$\ZZ[x_0, x_1, x_2, \cdots, x_{n+1}]$$ by this method.

EXAMPLES:

sage: P = posets.DiamondPoset(5)
sage: P.flag_f_polynomial()
3*x1*x2 + x2

sage: P = Poset({1: [2, 3], 2: , 3: , 4: , 5: })
sage: fl = P.flag_f_polynomial(); fl
2*x1*x2*x3 + 2*x1*x3 + 2*x2*x3 + x3
sage: q = polygen(ZZ,'q')
sage: fl(q,q,q,q) == P.f_polynomial()
True

sage: P = Poset({1: [2, 3, 4], 2: , 3: , 4: , 5: })
sage: P.flag_f_polynomial()
3*x1*x2*x3 + 3*x1*x3 + x2*x3 + x3

flag_h_polynomial()

Return the flag $$h$$-polynomial of the poset.

The poset is expected to be bounded and ranked.

If $$P$$ is a bounded ranked poset whose maximal element has rank $$n$$ (where the minimal element is set to have rank $$0$$), then the flag $$h$$-polynomial of $$P$$ is defined as the polynomial

$\prod_{k=1}^n (1-x_k) \cdot f \left(\frac{x_1}{1-x_1}, \frac{x_2}{1-x_2}, \cdots, \frac{x_n}{1-x_n}\right) \in \ZZ[x_1, x_2, \cdots, x_n],$

where $$f$$ is the flag $$f$$-polynomial of $$P$$ (see flag_f_polynomial()).

For technical reasons, the polynomial is returned in the slightly larger ring $$\QQ[x_0, x_1, x_2, \cdots, x_{n+1}]$$ by this method.

EXAMPLES:

sage: P = posets.DiamondPoset(5)
sage: P.flag_h_polynomial()
2*x1*x2 + x2

sage: P = Poset({1: [2, 3], 2: , 3: , 4: , 5: })
sage: fl = P.flag_h_polynomial(); fl
-x1*x2*x3 + x1*x3 + x2*x3 + x3
sage: q = polygen(ZZ,'q')
sage: fl(q,q,q,q) == P.h_polynomial()
True

sage: P = Poset({1: [2, 3, 4], 2: , 3: , 4: , 5: })
sage: P.flag_h_polynomial()
2*x1*x3 + x3

sage: P = posets.ChainPoset(4)
sage: P.flag_h_polynomial()
x3

frank_network()

Return Frank’s network of the poset.

This is defined in Section 8 of [BF1999].

OUTPUT:

A pair $$(G, e)$$, where $$G$$ is Frank’s network of $$P$$ encoded as a DiGraph, and $$e$$ is the cost function on its edges encoded as a dictionary (indexed by these edges, which in turn are encoded as tuples of 2 vertices).

Note

Frank’s network of $$P$$ is a certain directed graph with $$2|P| + 2$$ vertices, defined in Section 8 of [BF1999]. Its set of vertices consists of two vertices $$(0, p)$$ and $$(1, p)$$ for each element $$p$$ of $$P$$, as well as two vertices $$(-1, 0)$$ and $$(2, 0)$$. (These notations are not the ones used in [BF1999]; see the table below for their relation.) The edges are:

• for each $$p$$ in $$P$$, an edge from $$(-1, 0)$$ to $$(0, p)$$;
• for each $$p$$ in $$P$$, an edge from $$(1, p)$$ to $$(2, 0)$$;
• for each $$p$$ and $$q$$ in $$P$$ such that $$p \geq q$$, an edge from $$(0, p)$$ to $$(1, q)$$.

We make this digraph into a network in the sense of flow theory as follows: The vertex $$(-1, 0)$$ is considered as the source of this network, and the vertex $$(2, 0)$$ as the sink. The cost function is defined to be $$1$$ on the edge from $$(0, p)$$ to $$(1, p)$$ for each $$p \in P$$, and to be $$0$$ on every other edge. The capacity is $$1$$ on each edge. Here is how to translate this notations into that used in [BF1999]:

our notations                    [BF1999]
(-1, 0)                          s
(0, p)                          x_p
(1, p)                          y_p
(2, 0)                           t
a[e]                           a(e)


EXAMPLES:

sage: ps = [[16,12,14,-13],[[12,14],[14,-13],[12,16],[16,-13]]]
sage: G, e = Poset(ps).frank_network()
sage: G.edges(sort=True)
[((-1, 0), (0, -13), None), ((-1, 0), (0, 12), None), ((-1, 0), (0, 14), None), ((-1, 0), (0, 16), None), ((0, -13), (1, -13), None), ((0, -13), (1, 12), None), ((0, -13), (1, 14), None), ((0, -13), (1, 16), None), ((0, 12), (1, 12), None), ((0, 14), (1, 12), None), ((0, 14), (1, 14), None), ((0, 16), (1, 12), None), ((0, 16), (1, 16), None), ((1, -13), (2, 0), None), ((1, 12), (2, 0), None), ((1, 14), (2, 0), None), ((1, 16), (2, 0), None)]
sage: e
{((-1, 0), (0, -13)): 0,
((-1, 0), (0, 12)): 0,
((-1, 0), (0, 14)): 0,
((-1, 0), (0, 16)): 0,
((0, -13), (1, -13)): 1,
((0, -13), (1, 12)): 0,
((0, -13), (1, 14)): 0,
((0, -13), (1, 16)): 0,
((0, 12), (1, 12)): 1,
((0, 14), (1, 12)): 0,
((0, 14), (1, 14)): 1,
((0, 16), (1, 12)): 0,
((0, 16), (1, 16)): 1,
((1, -13), (2, 0)): 0,
((1, 12), (2, 0)): 0,
((1, 14), (2, 0)): 0,
((1, 16), (2, 0)): 0}
sage: qs = [[1,2,3,4,5,6,7,8,9],[[1,3],[3,4],[5,7],[1,9],[2,3]]]
sage: Poset(qs).frank_network()
(Digraph on 20 vertices,
{((-1, 0), (0, 1)): 0,
((-1, 0), (0, 2)): 0,
((-1, 0), (0, 3)): 0,
((-1, 0), (0, 4)): 0,
((-1, 0), (0, 5)): 0,
((-1, 0), (0, 6)): 0,
((-1, 0), (0, 7)): 0,
((-1, 0), (0, 8)): 0,
((-1, 0), (0, 9)): 0,
((0, 1), (1, 1)): 1,
((0, 2), (1, 2)): 1,
((0, 3), (1, 1)): 0,
((0, 3), (1, 2)): 0,
((0, 3), (1, 3)): 1,
((0, 4), (1, 1)): 0,
((0, 4), (1, 2)): 0,
((0, 4), (1, 3)): 0,
((0, 4), (1, 4)): 1,
((0, 5), (1, 5)): 1,
((0, 6), (1, 6)): 1,
((0, 7), (1, 5)): 0,
((0, 7), (1, 7)): 1,
((0, 8), (1, 8)): 1,
((0, 9), (1, 1)): 0,
((0, 9), (1, 9)): 1,
((1, 1), (2, 0)): 0,
((1, 2), (2, 0)): 0,
((1, 3), (2, 0)): 0,
((1, 4), (2, 0)): 0,
((1, 5), (2, 0)): 0,
((1, 6), (2, 0)): 0,
((1, 7), (2, 0)): 0,
((1, 8), (2, 0)): 0,
((1, 9), (2, 0)): 0})


AUTHOR:

• Darij Grinberg (2013-05-09)
ge(x, y)

Return True if $$x$$ is greater than or equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:, 4:[]})
sage: P.is_gequal(3, 1)
True
sage: P.is_gequal(2, 2)
True
sage: P.is_gequal(0, 1)
False

graphviz_string(graph_string='graph', edge_string='--')

Returns a representation in the DOT language, ready to render in graphviz.

EXAMPLES:

sage: P = Poset({'a':['b'],'b':['d'],'c':['d'],'d':['f'],'e':['f'],'f':[]})
sage: print(P.graphviz_string())
graph {
"f";"d";"b";"a";"c";"e";
"f"--"e";"d"--"c";"b"--"a";"d"--"b";"f"--"d";
}

greene_shape()

Return the Greene-Kleitman partition of self.

The Greene-Kleitman partition of a finite poset $$P$$ is the partition $$(c_1 - c_0, c_2 - c_1, c_3 - c_2, \ldots)$$, where $$c_k$$ is the maximum cardinality of a union of $$k$$ chains of $$P$$. Equivalently, this is the conjugate of the partition $$(a_1 - a_0, a_2 - a_1, a_3 - a_2, \ldots)$$, where $$a_k$$ is the maximum cardinality of a union of $$k$$ antichains of $$P$$.

See many sources, e. g., [BF1999], for proofs of this equivalence.

EXAMPLES:

sage: P = Poset([[3,2,1],[[3,1],[2,1]]])
sage: P.greene_shape()
[2, 1]
sage: P = Poset([[1,2,3,4],[[1,4],[2,4],[4,3]]])
sage: P.greene_shape()
[3, 1]
sage: P = Poset([[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22],[[1,4],[2,4],[4,3]]])
sage: P.greene_shape()
[3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
sage: P = Poset([[],[]])
sage: P.greene_shape()
[]


AUTHOR:

• Darij Grinberg (2013-05-09)
gt(x, y)

Return True if $$x$$ is greater than but not equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:, 4:[]})
sage: P.is_greater_than(3, 1)
True
sage: P.is_greater_than(1, 2)
False
sage: P.is_greater_than(3, 3)
False
sage: P.is_greater_than(0, 1)
False


For non-facade posets also > works:

sage: P = Poset({3: [1, 2]}, facade=False)
sage: P(2) > P(3)
True

h_polynomial()

Return the $$h$$-polynomial of a bounded poset self.

This is the $$h$$-polynomial of the order complex of the poset minus its bounds.

This is related to the $$f$$-polynomial by a simple change of variables:

$h(q) = (1-q)^{\deg f} f \left( \frac{q}{1-q} \right),$

where $$f$$ and $$h$$ denote the $$f$$-polynomial and the $$h$$-polynomial, respectively.

Warning

This is slightly different from the hPolynomial method in Macaulay2.

EXAMPLES:

sage: P = posets.AntichainPoset(3).order_ideals_lattice()
sage: P.h_polynomial()
q^3 + 4*q^2 + q
sage: P = posets.DiamondPoset(5)
sage: P.h_polynomial()
2*q^2 + q
sage: P = Poset({1: []})
sage: P.h_polynomial()
1

has_bottom()

Return True if the poset has a unique minimal element, and False otherwise.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:, 4:[]})
sage: P.has_bottom()
False
sage: Q = Poset({0:, 1:[]})
sage: Q.has_bottom()
True


has_isomorphic_subposet(other)

Return True if the poset contains a subposet isomorphic to other.

By subposet we mean that there exist a set X of elements such that self.subposet(X) is isomorphic to other.

INPUT:

• other – a finite poset

EXAMPLES:

sage: D = Poset({1:[2,3], 2:, 3:})
sage: T = Poset({1:[2,3], 2:[4,5], 3:[6,7]})
sage: N5 = posets.PentagonPoset()

sage: N5.has_isomorphic_subposet(T)
False
sage: N5.has_isomorphic_subposet(D)
True

sage: len([P for P in Posets(5) if P.has_isomorphic_subposet(D)])
11

has_top()

Return True if the poset has a unique maximal element, and False otherwise.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:[4, 5], 4:[], 5:[]})
sage: P.has_top()
False
sage: Q = Poset({0:, 1:, 2:, 3:, 4:[]})
sage: Q.has_top()
True


hasse_diagram()

Return the Hasse diagram of the poset as a Sage DiGraph.

The Hasse diagram is a directed graph where vertices are the elements of the poset and there is an edge from $$u$$ to $$v$$ whenever $$v$$ covers $$u$$ in the poset.

If dot2tex is installed, then this sets the Hasse diagram’s latex options to use the dot2tex formatting.

EXAMPLES:

sage: P = posets.DivisorLattice(12)
sage: H = P.hasse_diagram(); H
Digraph on 6 vertices
sage: P.cover_relations()
[[1, 2], [1, 3], [2, 4], [2, 6], [3, 6], [4, 12], [6, 12]]
sage: H.edges(labels=False)
[(1, 2), (1, 3), (2, 4), (2, 6), (3, 6), (4, 12), (6, 12)]

height(certificate=False)

Return the height (number of elements in a longest chain) of the poset.

INPUT:

• certificate – (default: False) whether to return a certificate

OUTPUT:

• If certificate=True return (h, c), where h is the height and c is a chain of maximum cardinality. If certificate=False return only the height.

EXAMPLES:

sage: P = Poset({0: , 2: [3, 4], 4: [5, 6]})
sage: P.height()
3
sage: posets.PentagonPoset().height(certificate=True)
(4, [0, 2, 3, 4])

incidence_algebra(R, prefix='I')

Return the incidence algebra of self over R.

OUTPUT:

EXAMPLES:

sage: P = posets.BooleanLattice(4)
sage: P.incidence_algebra(QQ)
Incidence algebra of Finite lattice containing 16 elements
over Rational Field

incomparability_graph()

Return the incomparability graph of the poset.

This is the complement of the comparability graph, i.e. an undirected graph where vertices are the elements of the poset and there is an edge between vertices if they are not comparable in the poset.

EXAMPLES:

sage: Y = Poset({1: , 2: [3, 4]})
sage: g = Y.incomparability_graph(); g
Incomparability graph on 4 vertices
sage: Y.compare_elements(1, 3) is not None
True
sage: g.has_edge(1, 3)
False

interval(x, y)

Return a list of the elements $$z$$ such that $$x \le z \le y$$.

INPUT:

• x – any element of the poset
• y – any element of the poset

EXAMPLES:

sage: uc = [[1,3,2],,[4,5,6],,,,,[]]
sage: dag = DiGraph(dict(zip(range(len(uc)),uc)))
sage: P = Poset(dag)
sage: I = set(map(P,[2,5,6,4,7]))
sage: I == set(P.interval(2,7))
True

sage: dg = DiGraph({"a":["b","c"], "b":["d"], "c":["d"]})
sage: P = Poset(dg, facade = False)
sage: P.interval("a","d")
[a, b, c, d]

intervals_number()

Return the number of relations in the poset.

A relation is a pair of elements $$x$$ and $$y$$ such that $$x\leq y$$ in the poset.

Relations are also often called intervals. The number of intervals is the dimension of the incidence algebra.

EXAMPLES:

sage: P = posets.PentagonPoset()
sage: P.relations_number()
13

sage: posets.TamariLattice(4).relations_number()
68

intervals_poset()

Return the natural partial order on the set of intervals of the poset.

OUTPUT:

a finite poset

The poset of intervals of a poset $$P$$ has the set of intervals $$[x,y]$$ in $$P$$ as elements, endowed with the order relation defined by $$[x_1,y_1] \leq [x_2,y_2]$$ if and only if $$x_1 \leq x_2$$ and $$y_1 \leq y_2$$.

This is also called $$P$$ to the power 2, meaning the poset of poset-morphisms from the 2-chain to $$P$$.

If $$P$$ is a lattice, the result is also a lattice.

EXAMPLES:

sage: P = Poset({0:})
sage: P.intervals_poset()
Finite poset containing 3 elements

sage: P = posets.PentagonPoset()
sage: P.intervals_poset()
Finite lattice containing 13 elements

is_EL_labelling(f, return_raising_chains=False)

Return True if f is an EL labelling of self.

A labelling $$f$$ of the edges of the Hasse diagram of a poset is called an EL labelling (edge lexicographic labelling) if for any two elements $$u$$ and $$v$$ with $$u \leq v$$,

• there is a unique $$f$$-raising chain from $$u$$ to $$v$$ in the Hasse diagram, and this chain is lexicographically first among all chains from $$u$$ to $$v$$.

For more details, see [Bj1980].

INPUT:

• f – a function taking two elements a and b in self such that b covers a and returning elements in a totally ordered set.
• return_raising_chains (optional; default:False) if True, returns the set of all raising chains in self, if possible.

EXAMPLES:

Let us consider a Boolean poset:

sage: P = Poset([[(0,0),(0,1),(1,0),(1,1)],[[(0,0),(0,1)],[(0,0),(1,0)],[(0,1),(1,1)],[(1,0),(1,1)]]],facade=True)
sage: label = lambda a,b: min( i for i in [0,1] if a[i] != b[i] )
sage: P.is_EL_labelling(label)
True
sage: P.is_EL_labelling(label,return_raising_chains=True)
{((0, 0), (0, 1)): ,
((0, 0), (1, 0)): ,
((0, 0), (1, 1)): [0, 1],
((0, 1), (1, 1)): ,
((1, 0), (1, 1)): }

is_antichain_of_poset(elms)

Return True if elms is an antichain of the poset and False otherwise.

Set of elements are an antichain of a poset if they are pairwise incomparable.

EXAMPLES:

sage: P = posets.BooleanLattice(5)
sage: P.is_antichain_of_poset([3, 5, 7])
False
sage: P.is_antichain_of_poset([3, 5, 14])
True

is_bounded()

Return True if the poset is bounded, and False otherwise.

A poset is bounded if it contains both a unique maximal element and a unique minimal element.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:[4, 5], 4:[], 5:[]})
sage: P.is_bounded()
False
sage: Q = posets.DiamondPoset(5)
sage: Q.is_bounded()
True

is_chain()

Return True if the poset is totally ordered (“chain”), and False otherwise.

EXAMPLES:

sage: I = Poset({0:, 1:, 2:, 3:})
sage: I.is_chain()
True

sage: II = Poset({0:, 2:})
sage: II.is_chain()
False

sage: V = Poset({0:[1, 2]})
sage: V.is_chain()
False

is_chain_of_poset(elms, ordered=False)

Return True if elms is a chain of the poset, and False otherwise.

Set of elements are a chain of a poset if they are comparable to each other.

INPUT:

• elms – a list or other iterable containing some elements of the poset
• ordered – a Boolean. If True, then return True only if elements in elms are strictly increasing in the poset; this makes no sense if elms is a set. If False (the default), then elements can be repeated and be in any order.

EXAMPLES:

sage: P = Poset((divisors(12), attrcall("divides")))
sage: sorted(P.list())
[1, 2, 3, 4, 6, 12]
sage: P.is_chain_of_poset([12, 3])
True
sage: P.is_chain_of_poset({3, 4, 12})
False
sage: P.is_chain_of_poset([12, 3], ordered=True)
False
sage: P.is_chain_of_poset((1, 1, 3))
True
sage: P.is_chain_of_poset((1, 1, 3), ordered=True)
False
sage: P.is_chain_of_poset((1, 3), ordered=True)
True

is_connected()

Return True if the poset is connected, and False otherwise.

A poset is connected if it’s Hasse diagram is connected.

If a poset is not connected, then it can be divided to parts $$S_1$$ and $$S_2$$ so that every element of $$S_1$$ is incomparable to every element of $$S_2$$.

EXAMPLES:

sage: P = Poset({1:[2, 3], 3:[4, 5]})
sage: P.is_connected()
True

sage: P = Poset({1:[2, 3], 3:[4, 5], 6:[7, 8]})
sage: P.is_connected()
False

is_eulerian(k=None, certificate=False)

Return True if the poset is Eulerian, and False otherwise.

The poset is expected to be graded and bounded.

A poset is Eulerian if every non-trivial interval has the same number of elements of even rank as of odd rank. A poset is $$k$$-eulerian if every non-trivial interval up to rank $$k$$ is Eulerian.

INPUT:

• k, an integer – only check if the poset is $$k$$-eulerian. If None (the default), check if the poset is Eulerian.
• certificate, a Boolean – (default: False) whether to return a certificate

OUTPUT:

• If certificate=True return either True, None or False, (a, b), where the interval (a, b) is not Eulerian. If certificate=False return True or False.

EXAMPLES:

sage: P = Poset({0: [1, 2, 3], 1: [4, 5], 2: [4, 6], 3: [5, 6],
....:            4: [7, 8], 5: [7, 8], 6: [7, 8], 7: , 8: })
sage: P.is_eulerian()
True
sage: P = Poset({0: [1, 2, 3], 1: [4, 5, 6], 2: [4, 6], 3: [5,6],
....:            4: , 5:, 6:})
sage: P.is_eulerian()
False


Canonical examples of Eulerian posets are the face lattices of convex polytopes:

sage: P = polytopes.cube().face_lattice()
sage: P.is_eulerian()
True


A poset that is 3- but not 4-eulerian:

sage: P = Poset(DiGraph('[email protected][email protected][email protected][email protected][email protected][email protected]??O??')); P
Finite poset containing 14 elements
sage: P.is_eulerian(k=3)
True
sage: P.is_eulerian(k=4)
False


Getting an interval that is not Eulerian:

sage: P = posets.DivisorLattice(12)
sage: P.is_eulerian(certificate=True)
(False, (1, 4))

is_gequal(x, y)

Return True if $$x$$ is greater than or equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:, 4:[]})
sage: P.is_gequal(3, 1)
True
sage: P.is_gequal(2, 2)
True
sage: P.is_gequal(0, 1)
False

is_graded()

Return True if the poset is graded, and False otherwise.

A poset is graded if all its maximal chains have the same length.

There are various competing definitions for graded posets (see Wikipedia article Graded_poset). This definition is from section 3.1 of Richard Stanley’s Enumerative Combinatorics, Vol. 1 [EnumComb1]. Some sources call these posets tiered.

Every graded poset is ranked. The converse is true for bounded posets, including lattices.

EXAMPLES:

sage: P = posets.PentagonPoset()  # Not even ranked
False

sage: P = Poset({1:[2, 3], 3:})  # Ranked, but not graded
False

sage: P = Poset({1:[3, 4], 2:[3, 4], 5:})
True

sage: P = Poset([, , [], , []])
False

is_greater_than(x, y)

Return True if $$x$$ is greater than but not equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:, 4:[]})
sage: P.is_greater_than(3, 1)
True
sage: P.is_greater_than(1, 2)
False
sage: P.is_greater_than(3, 3)
False
sage: P.is_greater_than(0, 1)
False


For non-facade posets also > works:

sage: P = Poset({3: [1, 2]}, facade=False)
sage: P(2) > P(3)
True

is_greedy(certificate=False)

Return True if the poset is greedy, and False otherwise.

A poset is greedy if every greedy linear extension has the same number of jumps.

INPUT:

• certificate – (default: False) whether to return a certificate

OUTPUT:

• If certificate=True return either (True, None) or (False, (A, B)) where $$A$$ and $$B$$ are greedy linear extension so that $$B$$ has more jumps. If certificate=False return True or False.

EXAMPLES:

This is not a self-dual property:

sage: W = Poset({1: [3, 4], 2: [4, 5]})
sage: M = W.dual()
sage: W.is_greedy()
True
sage: M.is_greedy()
False


Getting a certificate:

sage: N = Poset({1: , 2: [3, 4]})
sage: N.is_greedy(certificate=True)
(False, ([1, 2, 4, 3], [2, 4, 1, 3]))

is_incomparable_chain_free(m, n=None)

Return True if the poset is $$(m+n)$$-free, and False otherwise.

A poset is $$(m+n)$$-free if there is no incomparable chains of lengths $$m$$ and $$n$$. Three cases have special name (see [EnumComb1], exercise 3.15):

• ‘’interval order’’ is $$(2+2)$$-free
• ‘’semiorder’’ (or ‘’unit interval order’‘) is $$(1+3)$$-free and $$(2+2)$$-free
• ‘’weak order’’ is $$(1+2)$$-free.

INPUT:

• m, n - positive integers

It is also possible to give a list of integer pairs as argument. See below for an example.

EXAMPLES:

sage: B3 = posets.BooleanLattice(3)
sage: B3.is_incomparable_chain_free(1, 3)
True
sage: B3.is_incomparable_chain_free(2, 2)
False

sage: IP6 = posets.IntegerPartitions(6)
sage: IP6.is_incomparable_chain_free(1, 3)
False
sage: IP6.is_incomparable_chain_free(2, 2)
True


A list of pairs as an argument:

sage: B3.is_incomparable_chain_free([[1, 3], [2, 2]])
False


We show how to get an incomparable chain pair:

sage: P = posets.PentagonPoset()
sage: chains_1_2 = Poset({0:[], 1:})
sage: incomps = P.isomorphic_subposets(chains_1_2)
sage: sorted(incomps.list()), incomps.cover_relations()
([1, 2, 3], [[2, 3]])


AUTHOR:

• Eric Rowland (2013-05-28)
is_induced_subposet(other)

Return True if the poset is an induced subposet of other, and False otherwise.

A poset $$P$$ is an induced subposet of $$Q$$ if every element of $$P$$ is an element of $$Q$$, and $$x \le_P y$$ iff $$x \le_Q y$$. Note that “induced” here has somewhat different meaning compared to that of graphs.

INPUT:

• other, a poset.

Note

This method does not check whether the poset is a isomorphic (i.e., up to relabeling) subposet of other, but only if other directly contains the poset as an induced subposet. For isomorphic subposets see has_isomorphic_subposet().

EXAMPLES:

sage: P = Poset({1:[2, 3]})
sage: Q = Poset({1:[2, 4], 2:})
sage: P.is_induced_subposet(Q)
False
sage: R = Poset({0:, 1:[3, 4], 3:, 4:})
sage: P.is_induced_subposet(R)
True

is_isomorphic(other, **kwds)

Return True if both posets are isomorphic.

EXAMPLES:

sage: P = Poset(([1,2,3],[[1,3],[2,3]]))
sage: Q = Poset(([4,5,6],[[4,6],[5,6]]))
sage: P.is_isomorphic(Q)
True

is_join_semilattice(certificate=False)

Return True if the poset has a join operation, and False otherwise.

A join is the least upper bound for given elements, if it exists.

INPUT:

• certificate – (default: False) whether to return a certificate

OUTPUT:

• If certificate=True return either (True, None) or (False, (a, b)) where elements $$a$$ and $$b$$ have no least upper bound. If certificate=False return True or False.

EXAMPLES:

sage: P = Poset([[1,3,2], , [4,5,6], , , , , []])
sage: P.is_join_semilattice()
True

sage: P = Poset({1:[3, 4], 2:[3, 4], 3:, 4:})
sage: P.is_join_semilattice()
False
sage: P.is_join_semilattice(certificate=True)
(False, (2, 1))


is_jump_critical(certificate=False)

Return True if the poset is jump-critical, and False otherwise.

A poset $$P$$ is jump-critical if every proper subposet has smaller jump number.

INPUT:

• certificate – (default: False) whether to return a certificate

OUTPUT:

• If certificate=True return either (True, None) or (False, e) so that removing element $$e$$ from the poset does not decrease the jump number. If certificate=False return True or False.

EXAMPLES:

sage: P = Poset({1: [3, 6], 2: [3, 4, 5], 4: [6, 7], 5: })
sage: P.is_jump_critical()
True

sage: P = posets.PentagonPoset()
sage: P.is_jump_critical()
False
sage: P.is_jump_critical(certificate=True)
(False, 3)

is_lequal(x, y)

Return True if $$x$$ is less than or equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:, 4:[]})
sage: P.is_lequal(2, 4)
True
sage: P.is_lequal(2, 2)
True
sage: P.is_lequal(0, 1)
False
sage: P.is_lequal(3, 2)
False

is_less_than(x, y)

Return True if $$x$$ is less than but not equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:, 4:[]})
sage: P.is_less_than(1, 3)
True
sage: P.is_less_than(0, 1)
False
sage: P.is_less_than(2, 2)
False


For non-facade posets also < works:

sage: P = Poset({3: [1, 2]}, facade=False)
sage: P(1) < P(2)
False

is_linear_extension(l)

Returns whether l is a linear extension of self

INPUT:

• l – a list (or iterable) containing all of the elements of self exactly once

EXAMPLES:

sage: P = Poset((divisors(12), attrcall("divides")), facade=True, linear_extension=True)
sage: P.list()
[1, 2, 3, 4, 6, 12]
sage: P.is_linear_extension([1, 2, 4, 3, 6, 12])
True
sage: P.is_linear_extension([1, 2, 4, 6, 3, 12])
False

sage: [p for p in Permutations(list(P)) if P.is_linear_extension(p)]
[[1, 2, 3, 4, 6, 12],
[1, 2, 3, 6, 4, 12],
[1, 2, 4, 3, 6, 12],
[1, 3, 2, 4, 6, 12],
[1, 3, 2, 6, 4, 12]]
sage: list(P.linear_extensions())
[[1, 2, 3, 4, 6, 12],
[1, 2, 4, 3, 6, 12],
[1, 3, 2, 4, 6, 12],
[1, 3, 2, 6, 4, 12],
[1, 2, 3, 6, 4, 12]]


Note

This is used and systematically tested in LinearExtensionsOfPosets

is_meet_semilattice(certificate=False)

Return True if the poset has a meet operation, and False otherwise.

A meet is the greatest lower bound for given elements, if it exists.

INPUT:

• certificate – (default: False) whether to return a certificate

OUTPUT:

• If certificate=True return either (True, None) or (False, (a, b)) where elements $$a$$ and $$b$$ have no greatest lower bound. If certificate=False return True or False.

EXAMPLES:

sage: P = Poset({1:[2, 3, 4], 2:[5, 6], 3:, 4:[6, 7]})
sage: P.is_meet_semilattice()
True

sage: Q = P.dual()
sage: Q.is_meet_semilattice()
False

sage: V = posets.IntegerPartitions(5)
sage: V.is_meet_semilattice(certificate=True)
(False, ((2, 2, 1), (3, 1, 1)))


is_parent_of(x)

Return True if x is an element of the poset.

is_rank_symmetric()

Return True if the poset is rank symmetric, and False otherwise.

The poset is expected to be graded and connected.

A poset of rank $$h$$ (maximal chains have $$h+1$$ elements) is rank symmetric if the number of elements are equal in ranks $$i$$ and $$h-i$$ for every $$i$$ in $$0, 1, \ldots, h$$.

EXAMPLES:

sage: P = Poset({1:[3, 4, 5], 2:[3, 4, 5], 3:, 4:, 5:})
sage: P.is_rank_symmetric()
True
sage: P = Poset({1:, 2:[3, 4], 3:, 4:})
sage: P.is_rank_symmetric()
False

is_ranked()

Return True if the poset is ranked, and False otherwise.

A poset is ranked if there is a function $$r$$ from poset elements to integers so that $$r(x)=r(y)+1$$ when $$x$$ covers $$y$$.

Informally said a ranked poset can be “levelized”: every element is on a “level”, and every cover relation goes only one level up.

EXAMPLES:

sage: P = Poset( ([1, 2, 3, 4], [[1, 2], [2, 4], [3, 4]] ))
sage: P.is_ranked()
True

sage: P = Poset([[1, 5], [2, 6], , ,[], [6, 3], ])
sage: P.is_ranked()
False

is_series_parallel()

Return True if the poset is series-parallel, and False otherwise.

A poset is series-parallel if it can be built up from one-element posets using the operations of disjoint union and ordinal sum. This is also called N-free property: every poset that is not series-parallel contains a subposet isomorphic to the 4-element N-shaped poset where $$a > c, d$$ and $$b > d$$.

Note

Some papers use the term N-free for posets having no N-shaped poset as a cover-preserving subposet. This definition is not used here.

EXAMPLES:

sage: VA = Poset({1: [2, 3], 4: , 6: })
sage: VA.is_series_parallel()
True
sage: big_N = Poset({1: [2, 4], 2: , 4:, 5:, 6:})
sage: big_N.is_series_parallel()
False

is_slender(certificate=False)

Return True if the poset is slender, and False otherwise.

A finite graded poset is slender if every rank 2 interval contains three or four elements, as defined in [Stan2009]. (This notion of “slender” is unrelated to the eponymous notion defined by Graetzer and Kelly in “The Free $$\mathfrak{m}$$-Lattice on the Poset $$H$$”, Order 1 (1984), 47–65.)

This function does not check if the poset is graded or not. Instead it just returns True if the poset does not contain 5 distinct elements $$x$$, $$y$$, $$a$$, $$b$$ and $$c$$ such that $$x \lessdot a,b,c \lessdot y$$ where $$\lessdot$$ is the covering relation.

INPUT:

• certificate – (default: False) whether to return a certificate

OUTPUT:

• If certificate=True return either (True, None) or (False, (a, b)) so that the interval $$[a, b]$$ has at least five elements. If certificate=False return True or False.

EXAMPLES:

sage: P = Poset(([1, 2, 3, 4], [[1, 2], [1, 3], [2, 4], [3, 4]]))
sage: P.is_slender()
True
sage: P = Poset(([1,2,3,4,5],[[1,2],[1,3],[1,4],[2,5],[3,5],[4,5]]))
sage: P.is_slender()
False

sage: W = WeylGroup(['A', 2])
sage: G = W.bruhat_poset()
sage: G.is_slender()
True
sage: W = WeylGroup(['A', 3])
sage: G = W.bruhat_poset()
sage: G.is_slender()
True

sage: P = posets.IntegerPartitions(6)
sage: P.is_slender(certificate=True)
(False, ((6,), (3, 2, 1)))

is_sperner()

Return True if the poset is Sperner, and False otherwise.

The poset is expected to be ranked.

A poset is Sperner, if no antichain is larger than the largest rank level (one of the sets of elements of the same rank) in the poset.

EXAMPLES:

sage: posets.SetPartitions(3).is_sperner()
True

sage: P = Poset({0:[3,4,5],1:,2:})
sage: P.is_sperner()
False

isomorphic_subposets(other)

Return a list of subposets of self isomorphic to other.

By subposet we mean self.subposet(X) which is isomorphic to other and where X is a subset of elements of self.

INPUT:

• other – a finite poset

EXAMPLES:

sage: C2 = Poset({0:})
sage: C3 = Poset({'a':['b'], 'b':['c']})
sage: L = sorted(x.cover_relations() for x in C3.isomorphic_subposets(C2))
sage: for x in L: print(x)
[['a', 'b']]
[['a', 'c']]
[['b', 'c']]

sage: D = Poset({1:[2,3], 2:, 3:})
sage: N5 = posets.PentagonPoset()
sage: len(N5.isomorphic_subposets(D))
2


Note

If this function takes too much time, try using isomorphic_subposets_iterator().

isomorphic_subposets_iterator(other)

Return an iterator over the subposets of self isomorphic to other.

By subposet we mean self.subposet(X) which is isomorphic to other and where X is a subset of elements of self.

INPUT:

• other – a finite poset

EXAMPLES:

sage: D = Poset({1:[2,3], 2:, 3:})
sage: N5 = posets.PentagonPoset()
sage: for P in N5.isomorphic_subposets_iterator(D):
....:     print(P.cover_relations())
[[0, 1], [0, 2], [1, 4], [2, 4]]
[[0, 1], [0, 3], [1, 4], [3, 4]]
[[0, 1], [0, 2], [1, 4], [2, 4]]
[[0, 1], [0, 3], [1, 4], [3, 4]]


Warning

This function will return same subposet as many times as there are automorphism on it. This is due to subgraph_search_iterator() returning labelled subgraphs. On the other hand, this function does not eat memory like isomorphic_subposets() does.

jump_number(certificate=False)

Return the jump number of the poset.

A jump in a linear extension $$[e_1, \ldots, e_n]$$ of a poset $$P$$ is a pair $$(e_i, e_{i+1})$$ so that $$e_{i+1}$$ does not cover $$e_i$$ in $$P$$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.

INPUT:

• certificate – (default: False) Whether to return a certificate

OUTPUT:

• If certificate=True return a pair $$(n, l)$$ where $$n$$ is the jump number and $$l$$ is a linear extension with $$n$$ jumps. If certificate=False return only the jump number.

EXAMPLES:

sage: B3 = posets.BooleanLattice(3)
sage: B3.jump_number()
3

sage: N = Poset({1: [3, 4], 2: })
sage: N.jump_number(certificate=True)
(1, [1, 4, 2, 3])


ALGORITHM:

It is known that every poset has a greedy linear extension – an extension $$[e_1, e_2, \ldots, e_n]$$ where every $$e_{i+1}$$ is an upper cover of $$e_i$$ if that is possible – with the smallest possible number of jumps; see [Sys1987].

Hence it suffices to test only those. We do that by backtracking.

The problem is proven to be NP-complete.

kazhdan_lusztig_polynomial(x=None, y=None, q=None, canonical_labels=None)

Return the Kazhdan-Lusztig polynomial $$P_{x,y}(q)$$ of the poset.

The poset is expected to be ranked.

We follow the definition given in [EPW14]. Let $$G$$ denote a graded poset with unique minimal and maximal elements and $$\chi_G$$ denote the characteristic polynomial of $$G$$. Let $$I_x$$ and $$F^x$$ denote the principal order ideal and filter of $$x$$ respectively. Define the Kazhdan-Lusztig polynomial of $$G$$ as the unique polynomial $$P_G(q)$$ satisfying the following:

1. If $$\operatorname{rank} G = 0$$, then $$P_G(q) = 1$$.

2. If $$\operatorname{rank} G > 0$$, then $$\deg P_G(q) < \frac{1}{2} \operatorname{rank} G$$.

3. We have

$q^{\operatorname{rank} G} P_G(q^{-1}) = \sum_{x \in G} \chi_{I_x}(q) P_{F^x}(q).$

We then extend this to $$P_{x,y}(q)$$ by considering the subposet corresponding to the (closed) interval $$[x, y]$$. We also define $$P_{\emptyset}(q) = 0$$ (so if $$x \not\leq y$$, then $$P_{x,y}(q) = 0$$).

INPUT:

• q – (default: $$q \in \ZZ[q]$$) the indeterminate $$q$$
• x – (default: the minimal element) the element $$x$$
• y – (default: the maximal element) the element $$y$$
• canonical_labels – (optional) for subposets, use the canonical labeling (this can limit recursive calls for posets with large amounts of symmetry, but producing the labeling takes time); if not specified, this is True if x and y are both not specified and False otherwise

EXAMPLES:

sage: L = posets.BooleanLattice(3)
sage: L.kazhdan_lusztig_polynomial()
1

sage: L = posets.SymmetricGroupWeakOrderPoset(4)
sage: L.kazhdan_lusztig_polynomial()
1
sage: x = '2314'
sage: y = '3421'
sage: L.kazhdan_lusztig_polynomial(x, y)
-q + 1
sage: L.kazhdan_lusztig_polynomial(x, y, var('t'))
-t + 1


AUTHORS:

• Travis Scrimshaw (27-12-2014)
le(x, y)

Return True if $$x$$ is less than or equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:, 4:[]})
sage: P.is_lequal(2, 4)
True
sage: P.is_lequal(2, 2)
True
sage: P.is_lequal(0, 1)
False
sage: P.is_lequal(3, 2)
False

lequal_matrix(ring=Integer Ring, sparse=False)

Compute the matrix whose (i,j) entry is 1 if self.linear_extension()[i] < self.linear_extension()[j] and 0 otherwise.

INPUT:

• ring – the ring of coefficients (default: ZZ)
• sparse – whether the returned matrix is sparse or not (default: True)

EXAMPLES:

sage: P = Poset([[1,3,2],,[4,5,6],,,,,[]], facade = False)
sage: LEQM = P.lequal_matrix(); LEQM
[1 1 1 1 1 1 1 1]
[0 1 0 1 0 0 0 1]
[0 0 1 1 1 0 1 1]
[0 0 0 1 0 0 0 1]
[0 0 0 0 1 0 0 1]
[0 0 0 0 0 1 1 1]
[0 0 0 0 0 0 1 1]
[0 0 0 0 0 0 0 1]
sage: LEQM[1,3]
1
sage: P.linear_extension() < P.linear_extension()
True
sage: LEQM[2,5]
0
sage: P.linear_extension() < P.linear_extension()
False


We now demonstrate the usage of the optional parameters:

sage: P.lequal_matrix(ring=QQ, sparse=False).parent()
Full MatrixSpace of 8 by 8 dense matrices over Rational Field

level_sets()

Return elements grouped by maximal number of cover relations from a minimal element.

This returns a list of lists l such that l[i] is the set of minimal elements of the poset obtained by removing the elements in l, l, ..., l[i-1]. (In particular, l is the set of minimal elements of self.)

Every level is an antichain of the poset.

EXAMPLES:

sage: P = Poset({0:[1,2],1:,2:,3:[]})
sage: P.level_sets()
[, [1, 2], ]

sage: Q = Poset({0:[1,2], 1:, 2:, 3:})
sage: Q.level_sets()
[, [1, 2], , ]


dilworth_decomposition() to return elements grouped to chains.

lexicographic_sum(P)

Return the lexicographic sum using this poset as index.

In the lexicographic sum of posets $$P_t$$ by index poset $$T$$ we have $$x \le y$$ if either $$x \le y$$ in $$P_t$$ for some $$t \in T$$, or $$x \in P_i$$, $$y \in P_j$$ and $$i \le j$$ in $$T$$.

Informally said we substitute every element of $$T$$ by corresponding poset $$P_t$$.

Mathematically, it is only defined when all $$P_t$$ have no common element; here we force that by giving them different names in the resulting poset.

disjoint_union() and ordinal_sum() are special cases of lexicographic sum where the index poset is an (anti)chain. ordinal_product() is a special case where every $$P_t$$ is same poset.

INPUT:

• P – dictionary whose keys are elements of this poset, values are posets

EXAMPLES:

sage: N = Poset({1: [3, 4], 2: })
sage: P = {1: posets.PentagonPoset(), 2: N, 3: posets.ChainPoset(3), 4: posets.AntichainPoset(4)}
sage: NP = N.lexicographic_sum(P); NP
Finite poset containing 16 elements
sage: sorted(NP.minimal_elements())
[(1, 0), (2, 1), (2, 2)]

linear_extension(linear_extension=None, check=True)

Return a linear extension of this poset.

A linear extension of a finite poset $$P$$ of size $$n$$ is a total ordering $$\pi := \pi_0 \pi_1 \ldots \pi_{n-1}$$ of its elements such that $$i<j$$ whenever $$\pi_i < \pi_j$$ in the poset $$P$$.

INPUT:

• linear_extension – (default: None) a list of the elements of self
• check – a boolean (default: True); whether to check that linear_extension is indeed a linear extension of self.

EXAMPLES:

sage: P = Poset((divisors(15), attrcall("divides")), facade=True)


Without optional argument, the default linear extension of the poset is returned, as a plain list:

sage: P.linear_extension()
[1, 3, 5, 15]


Otherwise, a full-featured linear extension is constructed as an element of P.linear_extensions():

sage: l = P.linear_extension([1,5,3,15]); l
[1, 5, 3, 15]
sage: type(l)
<class 'sage.combinat.posets.linear_extensions.LinearExtensionsOfPoset_with_category.element_class'>
sage: l.parent()
The set of all linear extensions of Finite poset containing 4 elements


By default, the linear extension is checked for correctness:

sage: l = P.linear_extension([1,3,15,5])
Traceback (most recent call last):
...
ValueError: [1, 3, 15, 5] is not a linear extension of Finite poset containing 4 elements


This can be disabled (at your own risks!) with:

sage: P.linear_extension([1,3,15,5], check=False)
[1, 3, 15, 5]


Todo

• Is it acceptable to have those two features for a single method?
• In particular, we miss a short idiom to get the default linear extension
linear_extensions(facade=False)

Returns the enumerated set of all the linear extensions of this poset

INPUT:

• facade – a boolean (default: False); whether to return the linear extensions as plain lists

Warning

The facade option is not yet fully functional:

sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
The set of all linear extensions of Finite poset containing 6 elements with distinguished linear extension
sage: L([1, 2, 3, 4, 6, 12])
Traceback (most recent call last):
...
TypeError: Cannot convert list to sage.structure.element.Element


EXAMPLES:

sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
sage: P.list()
[1, 2, 3, 4, 6, 12]
sage: L = P.linear_extensions(); L
The set of all linear extensions of Finite poset containing 6 elements with distinguished linear extension
sage: l = L.an_element(); l
[1, 2, 3, 4, 6, 12]
sage: L.cardinality()
5
sage: L.list()
[[1, 2, 3, 4, 6, 12],
[1, 2, 4, 3, 6, 12],
[1, 3, 2, 4, 6, 12],
[1, 3, 2, 6, 4, 12],
[1, 2, 3, 6, 4, 12]]


Each element is aware that it is a linear extension of $$P$$:

sage: type(l.parent())
<class 'sage.combinat.posets.linear_extensions.LinearExtensionsOfPoset_with_category'>


With facade=True, the elements of L are plain lists instead:

sage: L = P.linear_extensions(facade=True)
sage: l = L.an_element()
sage: type(l)
<... 'list'>


Warning

In Sage <= 4.8, this function used to return a plain list of lists. To recover the previous functionality, please use:

sage: L = list(P.linear_extensions(facade=True)); L
[[1, 2, 3, 4, 6, 12],
[1, 2, 4, 3, 6, 12],
[1, 3, 2, 4, 6, 12],
[1, 3, 2, 6, 4, 12],
[1, 2, 3, 6, 4, 12]]
sage: type(L)
<... 'list'>

linear_extensions_graph()

Return the linear extensions graph of the poset.

Vertices of the graph are linear extensions of the poset. Two vertices are connected by an edge if the linear extensions differ by only one adjacent transposition.

EXAMPLES:

sage: N = Poset({1: [3, 4], 2: })
sage: G = N.linear_extensions_graph(); G
Graph on 5 vertices
sage: G.neighbors(N.linear_extension([1,2,3,4]))
[[2, 1, 3, 4], [1, 3, 2, 4], [1, 2, 4, 3]]

sage: chevron = Poset({1: [2, 6], 2: , 4: [3, 5], 6: })
sage: G = chevron.linear_extensions_graph(); G
Graph on 22 vertices
sage: G.size()
36

list()

List the elements of the poset. This just returns the result of linear_extension().

EXAMPLES:

sage: D = Poset({ 0:[1,2], 1:, 2:[3,4] }, facade = False)
sage: D.list()
[0, 1, 2, 3, 4]
sage: type(D.list())
<class 'sage.combinat.posets.posets.FinitePoset_with_category.element_class'>

lower_covers(x)

Return the list of lower covers of the element x.

A lower cover of $$x$$ is an element $$y$$ such that $$y < x$$ and there is no element $$z$$ so that $$y < z < x$$.

EXAMPLES:

sage: P = Poset([[1,5], [2,6], , , [], [6,3], ])
sage: P.lower_covers(3)
[2, 5]
sage: P.lower_covers(0)
[]

lower_covers_iterator(x)

Return an iterator over the lower covers of the element x.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:[]})
sage: l0 = P.lower_covers_iterator(3)
sage: type(l0)
<... 'generator'>
sage: next(l0)
2

lt(x, y)

Return True if $$x$$ is less than but not equal to $$y$$ in the poset, and False otherwise.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:, 4:[]})
sage: P.is_less_than(1, 3)
True
sage: P.is_less_than(0, 1)
False
sage: P.is_less_than(2, 2)
False


For non-facade posets also < works:

sage: P = Poset({3: [1, 2]}, facade=False)
sage: P(1) < P(2)
False

maximal_antichains()

Return the maximal antichains of the poset.

An antichain $$a$$ of poset $$P$$ is maximal if there is no element $$e \in P \setminus a$$ such that $$a \cup \{e\}$$ is an antichain.

EXAMPLES:

sage: P=Poset({'a':['b', 'c'], 'b':['d','e']})
sage: [sorted(anti) for anti in P.maximal_antichains()]
[['a'], ['b', 'c'], ['c', 'd', 'e']]

sage: posets.PentagonPoset().maximal_antichains()
[, [1, 2], [1, 3], ]

maximal_chains(partial=None)

Return all maximal chains of this poset.

Each chain is listed in increasing order.

INPUT:

• partial – list (optional); if present, find all maximal chains starting with the elements in partial

Returns list of the maximal chains of this poset.

This is used in constructing the order complex for the poset.

EXAMPLES:

sage: P = posets.BooleanLattice(3)
sage: P.maximal_chains()
[[0, 1, 3, 7], [0, 1, 5, 7], [0, 2, 3, 7], [0, 2, 6, 7], [0, 4, 5, 7], [0, 4, 6, 7]]
sage: P.maximal_chains(partial=[0,2])
[[0, 2, 3, 7], [0, 2, 6, 7]]
sage: Q = posets.ChainPoset(6)
sage: Q.maximal_chains()
[[0, 1, 2, 3, 4, 5]]

maximal_elements()

Return the list of the maximal elements of the poset.

EXAMPLES:

sage: P = Poset({0:,1:,2:,3:,4:[]})
sage: P.maximal_elements()



minimal_elements()

Return the list of the minimal elements of the poset.

EXAMPLES:

sage: P = Poset({0:,1:,2:,3:,4:[]})
sage: P(0) in P.minimal_elements()
True
sage: P(1) in P.minimal_elements()
True
sage: P(2) in P.minimal_elements()
True


moebius_function(x, y)

Return the value of the Möbius function of the poset on the elements x and y.

EXAMPLES:

sage: P = Poset([[1,2,3],,,,[]])
sage: P.moebius_function(P(0),P(4))
2
sage: sum(P.moebius_function(P(0),v) for v in P)
0
sage: sum(abs(P.moebius_function(P(0),v))
....:     for v in P)
6
sage: for u,v in P.cover_relations_iterator():
....:     if P.moebius_function(u,v) != -1:
....:         print("Bug in moebius_function!")

sage: Q = Poset([[1,3,2],,[4,5,6],,,,,[]])
sage: Q.moebius_function(Q(0),Q(7))
0
sage: Q.moebius_function(Q(0),Q(5))
0
sage: Q.moebius_function(Q(2),Q(7))
2
sage: Q.moebius_function(Q(3),Q(3))
1
sage: sum([Q.moebius_function(Q(0),v) for v in Q])
0

moebius_function_matrix(ring=Integer Ring, sparse=False)

Return a matrix whose (i,j) entry is the value of the Möbius function evaluated at self.linear_extension()[i] and self.linear_extension()[j].

INPUT:

• ring – the ring of coefficients (default: ZZ)
• sparse – whether the returned matrix is sparse or not (default: True)

EXAMPLES:

sage: P = Poset([[4,2,3],[],,,])
sage: x,y = (P.linear_extension(),P.linear_extension())
sage: P.moebius_function(x,y)
-1
sage: M = P.moebius_function_matrix(); M
[ 1 -1 -1 -1  2]
[ 0  1  0  0 -1]
[ 0  0  1  0 -1]
[ 0  0  0  1 -1]
[ 0  0  0  0  1]
sage: M[0,4]
2
sage: M[0,1]
-1


We now demonstrate the usage of the optional parameters:

sage: P.moebius_function_matrix(ring=QQ, sparse=False).parent()
Full MatrixSpace of 5 by 5 dense matrices over Rational Field

open_interval(x, y)

Return the list of elements $$z$$ such that $$x < z < y$$ in the poset.

EXAMPLES:

sage: P = Poset((divisors(1000), attrcall("divides")))
sage: P.open_interval(2, 100)
[4, 10, 20, 50]

order_complex(on_ints=False)

Return the order complex associated to this poset.

The order complex is the simplicial complex with vertices equal to the elements of the poset, and faces given by the chains.

INPUT:

• on_ints – a boolean (default: False)

OUTPUT:

an order complex of type SimplicialComplex

EXAMPLES:

sage: P = posets.BooleanLattice(3)
sage: S = P.order_complex(); S
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5, 6, 7) and 6 facets
sage: S.f_vector()
[1, 8, 19, 18, 6]
sage: S.homology()      # S is contractible
{0: 0, 1: 0, 2: 0, 3: 0}
sage: Q = P.subposet([1,2,3,4,5,6])
sage: Q.order_complex().homology()    # a circle
{0: 0, 1: Z}

sage: P = Poset((divisors(15), attrcall("divides")), facade = True)
sage: P.order_complex()
Simplicial complex with vertex set (1, 3, 5, 15) and facets {(1, 3, 15), (1, 5, 15)}


If on_ints, then the elements of the poset are labelled $$0,1,\dots$$ in the chain complex:

sage: P.order_complex(on_ints=True)
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 1, 3), (0, 2, 3)}

order_filter(elements)

Return the order filter generated by the elements of an iterable elements.

$$I$$ is an order filter if, for any $$x$$ in $$I$$ and $$y$$ such that $$y \ge x$$, then $$y$$ is in $$I$$. This is also called upper set or upset.

EXAMPLES:

sage: P = Poset((divisors(1000), attrcall("divides")))
sage: P.order_filter([20, 25])
[20, 40, 25, 50, 100, 200, 125, 250, 500, 1000]

order_ideal(elements)

Return the order ideal generated by the elements of an iterable elements.

$$I$$ is an order ideal if, for any $$x$$ in $$I$$ and $$y$$ such that $$y \le x$$, then $$y$$ is in $$I$$. This is also called lower set or downset.

EXAMPLES:

sage: P = Poset((divisors(1000), attrcall("divides")))
sage: P.order_ideal([20, 25])
[1, 2, 4, 5, 10, 20, 25]

order_ideal_plot(elements)

Return a plot of the order ideal generated by the elements of an iterable elements.

$$I$$ is an order ideal if, for any $$x$$ in $$I$$ and $$y$$ such that $$y \le x$$, then $$y$$ is in $$I$$. This is also called lower set or downset.

EXAMPLES:

sage: P = Poset((divisors(1000), attrcall("divides")))
sage: P.order_ideal_plot([20, 25])
Graphics object consisting of 41 graphics primitives

order_polynomial()

Return the order polynomial of the poset.

The order polynomial $$\Omega_P(q)$$ of a poset $$P$$ is defined as the unique polynomial $$S$$ such that for each integer $$m \geq 1$$, $$S(m)$$ is the number of order-preserving maps from $$P$$ to $$\{1,\ldots,m\}$$.

See sections 3.12 and 3.15 of [EnumComb1], and also [St1986].

EXAMPLES:

sage: P = posets.AntichainPoset(3)
sage: P.order_polynomial()
q^3

sage: P = posets.ChainPoset(3)
sage: f = P.order_polynomial(); f
1/6*q^3 + 1/2*q^2 + 1/3*q
sage: [f(i) for i in range(4)]
[0, 1, 4, 10]

order_polytope()

Return the order polytope of the poset self.

The order polytope of a finite poset $$P$$ is defined as the subset of $$\RR^P$$ consisting of all maps $$x : P \to \RR$$ satisfying

$0 \leq x(p) \leq 1 \mbox{ for all } p \in P,$

and

$x(p) \leq x(q) \mbox{ for all } p, q \in P \mbox{ satisfying } p < q.$

This polytope was defined and studied in [St1986].

EXAMPLES:

sage: P = posets.AntichainPoset(3)
sage: Q = P.order_polytope();Q
A 3-dimensional polyhedron in ZZ^3 defined as the convex hull of 8 vertices
sage: P = posets.PentagonPoset()
sage: Q = P.order_polytope();Q
A 5-dimensional polyhedron in ZZ^5 defined as the convex hull of 8 vertices

sage: P = Poset([[1,2,3],[[1,2],[1,3]]])
sage: Q = P.order_polytope()
sage: Q.contains((1,0,0))
False
sage: Q.contains((0,1,1))
True

ordinal_product(other, labels='pairs')

Return the ordinal product of self and other.

The ordinal product of two posets $$P$$ and $$Q$$ is a partial order on the Cartesian product of the underlying sets of $$P$$ and $$Q$$, defined as follows (see [EnumComb1], p. 284).

In the ordinal product, $$(p,q) \leq (p',q')$$ if either $$p \leq p'$$ or $$p = p'$$ and $$q \leq q'$$.

This construction is not symmetric in $$P$$ and $$Q$$. Informally said we put a copy of $$Q$$ in place of every element of $$P$$.

INPUT:

• other – a poset
• labels – either 'integers' or 'pairs' (default); how the resulting poset will be labeled

EXAMPLES:

sage: P1 = Poset((['a', 'b'], [['a', 'b']]))
sage: P2 = Poset((['c', 'd'], [['c', 'd']]))
sage: P = P1.ordinal_product(P2); P
Finite poset containing 4 elements
sage: sorted(P.cover_relations())
[[('a', 'c'), ('a', 'd')], [('a', 'd'), ('b', 'c')],
[('b', 'c'), ('b', 'd')]]

ordinal_sum(other, labels='pairs')

Return a poset or (semi)lattice isomorphic to ordinal sum of the poset with other.

The ordinal sum of $$P$$ and $$Q$$ is a poset that contains every element and relation from both $$P$$ and $$Q$$, and where every element of $$P$$ is smaller than any element of $$Q$$.

Mathematically, it is only defined when $$P$$ and $$Q$$ have no common element; here we force that by giving them different names in the resulting poset.

The ordinal sum on lattices is a lattice; resp. for meet- and join-semilattices.

INPUT:

• other, a poset.
• labels - (defaults to ‘pairs’) If set to ‘pairs’, each element v in this poset will be named (0,v) and each element u in other will be named (1,u) in the result. If set to ‘integers’, the elements of the result will be relabeled with consecutive integers.

EXAMPLES:

sage: P1 = Poset( ([1, 2, 3, 4], [[1, 2], [1, 3], [1, 4]]) )
sage: P2 = Poset( ([1, 2, 3,], [[2,1], [3,1]]) )
sage: P3 = P1.ordinal_sum(P2); P3
Finite poset containing 7 elements
sage: len(P1.maximal_elements())*len(P2.minimal_elements())
6
sage: len(P1.cover_relations()+P2.cover_relations())
5
sage: len(P3.cover_relations()) # Every element of P2 is greater than elements of P1.
11
sage: P3.list()  # random
[(0, 1), (0, 2), (0, 4), (0, 3), (1, 2), (1, 3), (1, 1)]
sage: P4 = P1.ordinal_sum(P2, labels='integers')
sage: P4.list()  # random
[0, 1, 2, 3, 5, 6, 4]


Return type depends on input types:

sage: P = Poset({1:}); P
Finite poset containing 2 elements
sage: JL = JoinSemilattice({1:}); JL
Finite join-semilattice containing 2 elements
sage: L = LatticePoset({1:}); L
Finite lattice containing 2 elements
sage: P.ordinal_sum(L)
Finite poset containing 4 elements
sage: L.ordinal_sum(JL)
Finite join-semilattice containing 4 elements
sage: L.ordinal_sum(L)
Finite lattice containing 4 elements

ordinal_summands()

Return the ordinal summands of the poset as subposets.

The ordinal summands of a poset $$P$$ is the longest list of non-empty subposets $$P_1, \ldots, P_n$$ whose ordinal sum is $$P$$. This decomposition is unique.

EXAMPLES:

sage: P = Poset({'a': ['c', 'd'], 'b': ['d'], 'c': ['x', 'y'],
....: 'd': ['x', 'y']})
sage: parts = P.ordinal_summands(); parts
[Finite poset containing 4 elements, Finite poset containing 2 elements]
sage: sorted(parts)
['a', 'b', 'c', 'd']
sage: Q = parts.ordinal_sum(parts)
sage: Q.is_isomorphic(P)
True


ALGORITHM:

Suppose that a poset $$P$$ is the ordinal sum of posets $$L$$ and $$U$$. Then $$P$$ contains maximal antichains $$l$$ and $$u$$ such that every element of $$u$$ covers every element of $$l$$; they correspond to maximal elements of $$L$$ and minimal elements of $$U$$.

We consider a linear extension $$x_1,\ldots,x_n$$ of the poset’s elements.

We keep track of the maximal elements of subposet induced by elements $$0,\ldots,x_i$$ and minimal elements of subposet induced by elements $$x_{i+1},\ldots,x_n$$, incrementing $$i$$ one by one. We then check if $$l$$ and $$u$$ fit the previous description.

p_partition_enumerator(tup, R, weights=None, check=False)

Return a $$P$$-partition enumerator of self.

Given a total order $$\prec$$ on the elements of a finite poset $$P$$ (the order of $$P$$ and the total order $$\prec$$ can be unrelated; in particular, the latter does not have to extend the former), a $$P$$-partition enumerator is the quasisymmetric function $$\sum_f \prod_{p \in P} x_{f(p)}$$, where the first sum is taken over all $$P$$-partitions $$f$$.

A $$P$$-partition is a function $$f : P \to \{1,2,3,...\}$$ satisfying the following properties for any two elements $$i$$ and $$j$$ of $$P$$ satisfying $$i <_P j$$:

• if $$i \prec j$$ then $$f(i) \leq f(j)$$,
• if $$j \prec i$$ then $$f(i) < f(j)$$.

The optional argument weights allows constructing a generalized (“weighted”) version of the $$P$$-partition enumerator. Namely, weights should be a dictionary whose keys are the elements of P. Then, the generalized $$P$$-partition enumerator corresponding to weights weights is $$\sum_f \prod_{p \in P} x_{f(p)}^{w(p)}$$, where the sum is again over all $$P$$-partitions $$f$$. Here, $$w(p)$$ is weights[p]. The classical $$P$$-partition enumerator is the particular case obtained when all $$p$$ satisfy $$w(p) = 1$$.

In the language of [Grinb2016a], the generalized $$P$$-partition enumerator is the quasisymmetric function $$\Gamma\left(\mathbf{E}, w\right)$$, where $$\mathbf{E}$$ is the special double poset $$(P, <_P, \prec)$$, and where $$w$$ is the dictionary weights (regarded as a function from $$P$$ to the positive integers).

INPUT:

• tup – the tuple containing all elements of $$P$$ (each of them exactly once), in the order dictated by the total order $$\prec$$
• R – a commutative ring
• weights – (optional) a dictionary of positive integers, indexed by elements of $$P$$; any missing item will be understood as $$1$$

OUTPUT:

The $$P$$-partition enumerator of self according to tup in the algebra $$QSym$$ of quasisymmetric functions over the base ring $$R$$.

EXAMPLES:

sage: P = Poset([[1,2,3,4],[[1,4],[2,4],[4,3]]])
sage: FP = P.p_partition_enumerator((3,1,2,4), QQ, check=True); FP
2*M[1, 1, 1, 1] + 2*M[1, 2, 1] + M[2, 1, 1] + M[3, 1]

sage: expansion = FP.expand(5)
sage: xs = expansion.parent().gens()
sage: expansion == sum([xs[a]*xs[b]*xs[c]*xs[d] for a in range(5) for b in range(5) for c in range(5) for d in range(5) if a <= b and c <= b and b < d])
True

sage: P = Poset([[],[]])
sage: FP = P.p_partition_enumerator((), QQ, check=True); FP
M[]


With the weights parameter:

sage: P = Poset([[1,2,3,4],[[1,4],[2,4],[4,3]]])
sage: FP = P.p_partition_enumerator((3,1,2,4), QQ, weights={1: 1, 2: 2, 3: 1, 4: 1}, check=True); FP
M[1, 2, 1, 1] + M[1, 3, 1] + M[2, 1, 1, 1] + M[2, 2, 1] + M[3, 1, 1] + M[4, 1]
sage: FP = P.p_partition_enumerator((3,1,2,4), QQ, weights={2: 2}, check=True); FP
M[1, 2, 1, 1] + M[1, 3, 1] + M[2, 1, 1, 1] + M[2, 2, 1] + M[3, 1, 1] + M[4, 1]

sage: P = Poset([['a','b','c'], [['a','b'], ['a','c']]])
sage: FP = P.p_partition_enumerator(('b','c','a'), QQ, weights={'a': 3, 'b': 5, 'c': 7}, check=True); FP
M[3, 5, 7] + M[3, 7, 5] + M[3, 12]

sage: P = Poset([['a','b','c'], [['a','c'], ['b','c']]])
sage: FP = P.p_partition_enumerator(('b','c','a'), QQ, weights={'a': 3, 'b': 5, 'c': 7}, check=True); FP
M[3, 5, 7] + M[3, 12] + M[5, 3, 7] + M[8, 7]
sage: FP = P.p_partition_enumerator(('a','b','c'), QQ, weights={'a': 3, 'b': 5, 'c': 7}, check=True); FP
M[3, 5, 7] + M[3, 12] + M[5, 3, 7] + M[5, 10] + M[8, 7] + M

plot(label_elements=True, element_labels=None, layout='acyclic', cover_labels=None, **kwds)

Return a Graphic object for the Hasse diagram of the poset.

If the poset is ranked, the plot uses the rank function for the heights of the elements.

INPUT:

• Options to change element look:
• element_colors - a dictionary where keys are colors and values are lists of elements
• element_color - a color for elements not set in element_colors
• element_shape - the shape of elements, like 's' for square; see https://matplotlib.org/api/markers_api.html for the list
• element_size (default: 200) - the size of elements
• label_elements (default: True) - whether to display element labels
• element_labels (default: None) - a dictionary where keys are elements and values are labels to show
• Options to change cover relation look:
• cover_colors - a dictionary where keys are colors and values are lists of cover relations given as pairs of elements
• cover_color - a color for elements not set in cover_colors
• cover_style - style for cover relations: 'solid', 'dashed', 'dotted' or 'dashdot'
• cover_labels - a dictionary, list or function representing labels of the covers of the poset. When set to None (default) no label is displayed on the edges of the Hasse Diagram.
• cover_labels_background - a background color for cover relations. The default is “white”. To achieve a transparent background use “transparent”.
• Options to change overall look:
• figsize (default: 8) - size of the whole plot
• title - a title for the plot
• fontsize - fontsize for the title
• border (default: False) - whether to draw a border over the plot

Note

All options of GenericGraph.plot are also available through this function.

EXAMPLES:

This function can be used without any parameters:

sage: D12 = posets.DivisorLattice(12)
sage: D12.plot()
Graphics object consisting of 14 graphics primitives


Just the abstract form of the poset; examples of relabeling:

sage: D12.plot(label_elements=False)
Graphics object consisting of 8 graphics primitives
sage: d = {1: 0, 2: 'a', 3: 'b', 4: 'c', 6: 'd', 12: 1}
sage: D12.plot(element_labels=d)
Graphics object consisting of 14 graphics primitives
sage: d = {i:str(factor(i)) for i in D12}
sage: D12.plot(element_labels=d)
Graphics object consisting of 14 graphics primitives


Some settings for coverings:

sage: d = {(a, b): b/a for a, b in D12.cover_relations()}
sage: D12.plot(cover_labels=d, cover_color='gray', cover_style='dotted')
Graphics object consisting of 21 graphics primitives


To emphasize some elements and show some options:

sage: L = LatticePoset({0: [1, 2, 3, 4], 1: , 2: [6, 7],
....:                   3: [5, 9], 4: [5, 6, 10, 11], 5: ,
....:                   6: , 7: [12, 8, 9], 8: , 9: ,
....:                   10: , 11: , 12: })
sage: F = L.frattini_sublattice()
sage: F_internal = [c for c in F.cover_relations() if c in L.cover_relations()]
sage: L.plot(figsize=12, border=True, element_shape='s',
....:        element_size=400, element_color='white',
....:        element_colors={'blue': F, 'green': L.double_irreducibles()},
....:        cover_color='lightgray', cover_colors={'black': F_internal},
....:        title='The Frattini\nsublattice in blue', fontsize=10)
Graphics object consisting of 39 graphics primitives

product(other)

Return the Cartesian product of the poset with other.

The Cartesian (or ‘direct’) product of $$P$$ and $$Q$$ is defined by $$(p, q) \le (p', q')$$ iff $$p \le p'$$ in $$P$$ and $$q \le q'$$ in $$Q$$.

Product of (semi)lattices are returned as a (semi)lattice.

EXAMPLES:

sage: P = posets.ChainPoset(3)
sage: Q = posets.ChainPoset(4)
sage: PQ = P.product(Q) ; PQ
Finite lattice containing 12 elements
sage: len(PQ.cover_relations())
17
sage: Q.product(P).is_isomorphic(PQ)
True

sage: P = posets.BooleanLattice(2)
sage: Q = P.product(P)
sage: Q.is_isomorphic(posets.BooleanLattice(4))
True


One can also simply use $$*$$:

sage: P = posets.ChainPoset(2)
sage: Q = posets.ChainPoset(3)
sage: P*Q
Finite lattice containing 6 elements

promotion(i=1)

Compute the (extended) promotion on the linear extension of the poset self.

INPUT:

• i – an integer between $$1$$ and $$n$$ (default: $$1$$)

OUTPUT:

• an isomorphic poset, with the same default linear extension

The extended promotion is defined on a poset self of size $$n$$ by applying the promotion operator $$\tau_i \tau_{i+1} \cdots \tau_{n-1}$$ to the default linear extension $$\pi$$ of self (see promotion()), and relabeling self accordingly. For more details see [Stan2009].

When the elements of the poset self are labelled by $$\{1,2,\ldots,n\}$$, the linear extension is the identity, and $$i=1$$, the above algorithm corresponds to the promotion operator on posets defined by Schützenberger as follows. Remove $$1$$ from self and replace it by the minimum $$j$$ of all labels covering $$1$$ in the poset. Then, remove $$j$$ and replace it by the minimum of all labels covering $$j$$, and so on. This process ends when a label is a local maximum. Place the label $$n+1$$ at this vertex. Finally, decrease all labels by $$1$$.

EXAMPLES:

sage: P = Poset(([1,2], [[1,2]]), linear_extension=True, facade=False)
sage: P.promotion()
Finite poset containing 2 elements with distinguished linear extension
sage: P == P.promotion()
True

sage: P = Poset(([1,2,3,4,5,6,7], [[1,2],[1,4],[2,3],[2,5],[3,6],[4,7],[5,6]]))
sage: P.list()
[1, 2, 3, 5, 6, 4, 7]
sage: Q = P.promotion(4); Q
Finite poset containing 7 elements with distinguished linear extension
sage: Q.cover_relations()
[[1, 2], [1, 6], [2, 3], [2, 5], [3, 7], [5, 7], [6, 4]]


Note that if one wants to obtain the promotion defined by Schützenberger’s algorithm directly on the poset, one needs to make sure the linear extension is the identity:

sage: P = P.with_linear_extension([1,2,3,4,5,6,7])
sage: P.list()
[1, 2, 3, 4, 5, 6, 7]
sage: Q = P.promotion(4); Q
Finite poset containing 7 elements with distinguished linear extension
sage: Q.cover_relations()
[[1, 2], [1, 6], [2, 3], [2, 4], [3, 5], [4, 5], [6, 7]]
sage: Q = P.promotion()
sage: Q.cover_relations()
[[1, 2], [1, 3], [2, 4], [2, 5], [3, 6], [4, 7], [5, 7]]


Here is an example for a poset not labelled by $$\{1, 2, \ldots, n\}$$:

sage: P = Poset((divisors(30), attrcall("divides")), linear_extension=True)
sage: P.list()
[1, 2, 3, 5, 6, 10, 15, 30]
sage: P.cover_relations()
[[1, 2], [1, 3], [1, 5], [2, 6], [2, 10], [3, 6], [3, 15],
[5, 10], [5, 15], [6, 30], [10, 30], [15, 30]]
sage: Q = P.promotion(4); Q
Finite poset containing 8 elements with distinguished linear extension
sage: Q.cover_relations()
[[1, 2], [1, 3], [1, 6], [2, 5], [2, 15], [3, 5], [3, 10],
[5, 30], [6, 10], [6, 15], [10, 30], [15, 30]]


AUTHOR:

• Anne Schilling (2012-02-18)
random_linear_extension()

Return a random linear extension of the poset.

The distribution is not uniform.

EXAMPLES:

sage: set_random_seed(0)  # results are reproduceable
sage: P = posets.BooleanLattice(4)
sage: P.random_linear_extension()
[0, 2, 8, 1, 9, 4, 5, 10, 6, 12, 14, 13, 3, 7, 11, 15]

random_maximal_antichain()

Return a random maximal antichain of the poset.

The distribution is not uniform.

EXAMPLES:

sage: set_random_seed(0)  # results are reproduceable
sage: P = posets.BooleanLattice(4)
sage: P.random_maximal_antichain()
[1, 8, 2, 4]

random_maximal_chain()

Return a random maximal chain of the poset.

The distribution is not uniform.

EXAMPLES:

sage: set_random_seed(0)  # results are reproduceable
sage: P = posets.BooleanLattice(4)
sage: P.random_maximal_chain()
[0, 2, 10, 11, 15]

random_order_ideal(direction='down')

Return a random order ideal with uniform probability.

INPUT:

• direction'up', 'down' or 'antichain' (default: 'down')

OUTPUT:

A randomly selected order ideal (or order filter if direction='up', or antichain if direction='antichain') where all order ideals have equal probability of occurring.

ALGORITHM:

Uses the coupling from the past algorithm described in [Propp1997].

EXAMPLES:

sage: P = posets.BooleanLattice(3)
sage: P.random_order_ideal()
[0, 1, 2, 3, 4, 5, 6]
sage: P.random_order_ideal(direction='up')
[6, 7]
sage: P.random_order_ideal(direction='antichain')
[1, 2]

sage: P = posets.TamariLattice(5)
sage: a = P.random_order_ideal('antichain')
sage: P.is_antichain_of_poset(a)
True
sage: a = P.random_order_ideal('up')
sage: P.is_order_filter(a)
True
sage: a = P.random_order_ideal('down')
sage: P.is_order_ideal(a)
True

random_subposet(p)

Return a random subposet that contains each element with probability p.

EXAMPLES:

sage: P = posets.BooleanLattice(3)
sage: set_random_seed(0)  # Results are reproducible
sage: Q = P.random_subposet(0.5)
sage: Q.cover_relations()
[[0, 2], [0, 5], [2, 3], [3, 7], [5, 7]]

rank(element=None)

Return the rank of an element element in the poset self, or the rank of the poset if element is None.

(The rank of a poset is the length of the longest chain of elements of the poset. This is sometimes called the length of a poset.)

EXAMPLES:

sage: P = Poset([[1,3,2],,[4,5,6],,,,,[]], facade = False)
sage: P.rank(5)
2
sage: P.rank()
3
sage: Q = Poset([[1,2],,[],[]])

sage: P = posets.SymmetricGroupBruhatOrderPoset(4)
sage: [(v,P.rank(v)) for v in P]
[('1234', 0),
('1243', 1),
...
('4312', 5),
('4321', 6)]

rank_function()

Return the (normalized) rank function of the poset, if it exists.

A rank function of a poset $$P$$ is a function $$r$$ that maps elements of $$P$$ to integers and satisfies: $$r(x) = r(y) + 1$$ if $$x$$ covers $$y$$. The function $$r$$ is normalized such that its minimum value on every connected component of the Hasse diagram of $$P$$ is $$0$$. This determines the function $$r$$ uniquely (when it exists).

OUTPUT:

• a lambda function, if the poset admits a rank function
• None, if the poset does not admit a rank function

EXAMPLES:

sage: P = Poset(([1,2,3,4],[[1,4],[2,3],[3,4]]), facade=True)
sage: P.rank_function() is not None
True
sage: P.rank_function() is not None
False
sage: f = P.rank_function(); f is not None
True
sage: f(5)
0
sage: f(2)
0

rees_product(other)

Return the Rees product of self and other.

This is only defined if both posets are graded.

The underlying set is the set of pairs $$(p,q)$$ in the Cartesian product such that $$\operatorname{rk}(p) \geq \operatorname{rk}(q)$$.

This operation was defined by Björner and Welker in [BjWe2005]. Other references are [MBRe2011] and [LSW2012].

EXAMPLES:

sage: B3 = posets.BooleanLattice(3)
sage: B3t = B3.subposet(list(range(1,8)))
sage: C3 = posets.ChainPoset(3)
sage: D = B3t.rees_product(C3); D
Finite poset containing 12 elements
sage: sorted(D.minimal_elements())
[(1, 0), (2, 0), (4, 0)]
sage: sorted(D.maximal_elements())
[(7, 0), (7, 1), (7, 2)]
sage: D.with_bounds().moebius_function('bottom','top')
2

relabel(relabeling=None)

Return a copy of this poset with its elements relabeled.

INPUT:

• relabeling – a function, dictionary, list or tuple

The given function or dictionary must map each (non-wrapped) element of self to some distinct object. The given list or tuple must be made of distinct objects.

When the input is a list or a tuple, the relabeling uses the total ordering of the elements of the poset given by list(self).

If no relabeling is given, the poset is relabeled by integers from $$0$$ to $$n-1$$ according to one of its linear extensions. This means that $$i<j$$ as integers whenever $$i<j$$ in the relabeled poset.

EXAMPLES:

Relabeling using a function:

sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
sage: P.list()
[1, 2, 3, 4, 6, 12]
sage: P.cover_relations()
[[1, 2], [1, 3], [2, 4], [2, 6], [3, 6], [4, 12], [6, 12]]
sage: Q = P.relabel(lambda x: x+1)
sage: Q.list()
[2, 3, 4, 5, 7, 13]
sage: Q.cover_relations()
[[2, 3], [2, 4], [3, 5], [3, 7], [4, 7], [5, 13], [7, 13]]


Relabeling using a dictionary:

sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True, facade=False)
sage: relabeling = {c.element:i for (i,c) in enumerate(P)}
sage: relabeling
{1: 0, 2: 1, 3: 2, 4: 3, 6: 4, 12: 5}
sage: Q = P.relabel(relabeling)
sage: Q.list()
[0, 1, 2, 3, 4, 5]
sage: Q.cover_relations()
[[0, 1], [0, 2], [1, 3], [1, 4], [2, 4], [3, 5], [4, 5]]


Mind the c.element; this is because the relabeling is applied to the elements of the poset without the wrapping. Thanks to this convention, the same relabeling function can be used both for facade or non facade posets.

Relabeling using a list:

sage: P = posets.PentagonPoset()
sage: list(P)
[0, 1, 2, 3, 4]
sage: P.cover_relations()
[[0, 1], [0, 2], [1, 4], [2, 3], [3, 4]]
sage: Q = P.relabel(list('abcde'))
sage: Q.cover_relations()
[['a', 'b'], ['a', 'c'], ['b', 'e'], ['c', 'd'], ['d', 'e']]


Default behaviour is increasing relabeling:

sage: a2 = posets.ChainPoset(2)
sage: P = a2 * a2
sage: Q = P.relabel()
sage: Q.cover_relations()
[[0, 1], [0, 2], [1, 3], [2, 3]]


Relabeling a (semi)lattice gives a (semi)lattice:

sage: P = JoinSemilattice({0: })
sage: P.relabel(lambda n: n+1)
Finite join-semilattice containing 2 elements


Note

As can be seen in the above examples, the default linear extension of Q is that of P after relabeling. In particular, P and Q share the same internal Hasse diagram.

relations()

Return the list of all relations of the poset.

A relation is a pair of elements $$x$$ and $$y$$ such that $$x \leq y$$ in the poset.

The number of relations is the dimension of the incidence algebra.

OUTPUT:

A list of pairs (each pair is a list), where the first element of the pair is less than or equal to the second element.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:, 4:[]})
sage: P.relations()
[[1, 1], [1, 2], [1, 3], [1, 4], [0, 0], [0, 2], [0, 3],
[0, 4], [2, 2], [2, 3], [2, 4], [3, 3], [3, 4], [4, 4]]


AUTHOR:

• Rob Beezer (2011-05-04)
relations_iterator(strict=False)

Return an iterator for all the relations of the poset.

A relation is a pair of elements $$x$$ and $$y$$ such that $$x \leq y$$ in the poset.

INPUT:

• strict – a boolean (default False) if True, returns an iterator over relations $$x < y$$, excluding all relations $$x \leq x$$.

OUTPUT:

A generator that produces pairs (each pair is a list), where the first element of the pair is less than or equal to the second element.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:, 4:[]})
sage: it = P.relations_iterator()
sage: type(it)
<... 'generator'>
sage: next(it), next(it)
([1, 1], [1, 2])

sage: P = posets.PentagonPoset()
sage: list(P.relations_iterator(strict=True))
[[0, 1], [0, 2], [0, 4], [0, 3], [1, 4], [2, 3], [2, 4], [3, 4]]


AUTHOR:

• Rob Beezer (2011-05-04)
relations_number()

Return the number of relations in the poset.

A relation is a pair of elements $$x$$ and $$y$$ such that $$x\leq y$$ in the poset.

Relations are also often called intervals. The number of intervals is the dimension of the incidence algebra.

EXAMPLES:

sage: P = posets.PentagonPoset()
sage: P.relations_number()
13

sage: posets.TamariLattice(4).relations_number()
68

show(label_elements=True, element_labels=None, cover_labels=None, **kwds)

Displays the Hasse diagram of the poset.

INPUT:

• label_elements (default: True) - whether to display element labels
• element_labels (default: None) - a dictionary of element labels
• cover_labels - a dictionary, list or function representing labels of the covers of self. When set to None (default) no label is displayed on the edges of the Hasse Diagram.

Note

This method also accepts:

EXAMPLES:

sage: D = Poset({ 0:[1,2], 1:, 2:[3,4] })
sage: D.plot(label_elements=False)
Graphics object consisting of 6 graphics primitives
sage: D.show()
sage: elm_labs = {0:'a', 1:'b', 2:'c', 3:'d', 4:'e'}
sage: D.show(element_labels=elm_labs)


One more example with cover labels:

sage: P = posets.PentagonPoset()
sage: P.show(cover_labels=lambda a, b: a - b)

sorted(l, allow_incomparable=True, remove_duplicates=False)

Return the list $$l$$ sorted by the poset.

INPUT:

• l – a list of elements of the poset
• allow_incomparable – a Boolean. If True (the default), return incomparable elements in some order; if False, raise an error if l is not a chain of the poset.
• remove_duplicates - a Boolean. If True, remove duplicates from the output list.

EXAMPLES:

sage: P = posets.DivisorLattice(36)
sage: P.sorted([1, 4, 1, 6, 2, 12])  # Random order for 4 and 6
[1, 1, 2, 4, 6, 12]
sage: P.sorted([1, 4, 1, 6, 2, 12], remove_duplicates=True)
[1, 2, 4, 6, 12]
sage: P.sorted([1, 4, 1, 6, 2, 12], allow_incomparable=False)
Traceback (most recent call last):
...
ValueError: the list contains incomparable elements

sage: P = Poset({7:[1, 5], 1:[2, 6], 5:, 6:[3, 4]})
sage: P.sorted([4, 1, 4, 5, 7])  # Random order for 1 and 5
[7, 1, 5, 4, 4]
sage: P.sorted([1, 4, 4, 7], remove_duplicates=True)
[7, 1, 4]
sage: P.sorted([4, 1, 4, 5, 7], allow_incomparable=False)
Traceback (most recent call last):
...
ValueError: the list contains incomparable elements

star_product(other, labels='pairs')

Return a poset isomorphic to the star product of the poset with other.

Both this poset and other are expected to be bounded and have at least two elements.

Let $$P$$ be a poset with top element $$\top_P$$ and $$Q$$ be a poset with bottom element $$\bot_Q$$. The star product of $$P$$ and $$Q$$ is the ordinal sum of $$P \setminus \top_P$$ and $$Q \setminus \bot_Q$$.

Mathematically, it is only defined when $$P$$ and $$Q$$ have no common elements; here we force that by giving them different names in the resulting poset.

INPUT:

• other – a poset.
• labels – (defaults to ‘pairs’) If set to ‘pairs’, each element v in this poset will be named (0, v) and each element u in other will be named (1, u) in the result. If set to ‘integers’, the elements of the result will be relabeled with consecutive integers.

EXAMPLES:

This is mostly used to combine two Eulerian posets to third one, and makes sense for graded posets only:

sage: B2 = posets.BooleanLattice(2)
sage: B3 = posets.BooleanLattice(3)
sage: P = B2.star_product(B3); P
Finite poset containing 10 elements
sage: P.is_eulerian()
True


We can get elements as pairs or as integers:

sage: ABC = Poset({'a': ['b'], 'b': ['c']})
sage: XYZ = Poset({'x': ['y'], 'y': ['z']})
sage: ABC.star_product(XYZ).list()
[(0, 'a'), (0, 'b'), (1, 'y'), (1, 'z')]
sage: sorted(ABC.star_product(XYZ, labels='integers'))
[0, 1, 2, 3]

subposet(elements)

Return the poset containing given elements with partial order induced by this poset.

EXAMPLES:

sage: P = Poset({'a': ['c', 'd'], 'b': ['d','e'], 'c': ['f'],
....:            'd': ['f'], 'e': ['f']})
sage: Q = P.subposet(['a', 'b', 'f']); Q
Finite poset containing 3 elements
sage: Q.cover_relations()
[['b', 'f'], ['a', 'f']]


sage: P = posets.PentagonPoset(facade=False)
sage: Q = P.subposet([0, 1, 2, 4])
sage: Q(1) < Q(2)
False

top()

Return the unique maximal element of the poset, if it exists.

EXAMPLES:

sage: P = Poset({0:,1:,2:,3:[4,5],4:[],5:[]})
sage: P.top() is None
True
sage: Q = Poset({0:,1:[]})
sage: Q.top()
1


unwrap(element)

Return the element element of the poset self in unwrapped form.

INPUT:

• element – an element of self

EXAMPLES:

sage: P = Poset((divisors(15), attrcall("divides")), facade = False)
sage: x = P.an_element(); x
1
sage: x.parent()
Finite poset containing 4 elements
sage: P.unwrap(x)
1
sage: P.unwrap(x).parent()
Integer Ring


For a non facade poset, this is equivalent to using the .element attribute:

sage: P.unwrap(x) is x.element
True


For a facade poset, this does nothing:

sage: P = Poset((divisors(15), attrcall("divides")), facade=True)
sage: x = P.an_element()
sage: P.unwrap(x) is x
True


This method is useful in code where we don’t know if P is a facade poset or not.

upper_covers(x)

Return the list of upper covers of the element x.

An upper cover of $$x$$ is an element $$y$$ such that $$x < y$$ and there is no element $$z$$ so that $$x < z < y$$.

EXAMPLES:

sage: P = Poset([[1,5], [2,6], , , [], [6,3], ])
sage: P.upper_covers(1)
[2, 6]

upper_covers_iterator(x)

Return an iterator over the upper covers of the element x.

EXAMPLES:

sage: P = Poset({0:, 1:, 2:, 3:[]})
sage: type(P.upper_covers_iterator(0))
<... 'generator'>

width(certificate=False)

Return the width of the poset (the size of its longest antichain).

It is computed through a matching in a bipartite graph; see Wikipedia article Dilworth’s_theorem for more information. The width is also called Dilworth number.

INPUT:

• certificate – (default: False) whether to return a certificate

OUTPUT:

• If certificate=True return (w, a), where $$w$$ is the width of a poset and $$a$$ is an antichain of maximum cardinality. If certificate=False return only width of the poset.

EXAMPLES:

sage: P = posets.BooleanLattice(4)
sage: P.width()
6

sage: w, max_achain = P.width(certificate=True)
sage: sorted(max_achain)
[3, 5, 6, 9, 10, 12]

with_bounds(labels=('bottom', 'top'))

Return the poset with bottom and top elements adjoined.

This function adds top and bottom elements to the poset. It will always add elements, it does not check if the poset already has a bottom or a top element.

For lattices and semilattices this function returns a lattice.

INPUT:

• labels – A pair of elements to use as a bottom and top element of the poset. Default is strings 'bottom' and 'top'. Either of them can be None, and then a new bottom or top element will not be added.

EXAMPLES:

sage: V = Poset({0: [1, 2]})
sage: trafficsign = V.with_bounds(); trafficsign
Finite poset containing 5 elements
sage: trafficsign.list()
['bottom', 0, 1, 2, 'top']
sage: trafficsign = V.with_bounds(labels=(-1, -2))
sage: trafficsign.cover_relations()
[[-1, 0], [0, 1], [0, 2], [1, -2], [2, -2]]

sage: Y = V.with_bounds(labels=(-1, None))
sage: Y.cover_relations()
[[-1, 0], [0, 1], [0, 2]]

sage: P = posets.PentagonPoset()  # A lattice
sage: P.with_bounds()
Finite lattice containing 7 elements

sage: P.with_bounds()
Finite lattice containing 7 elements


without_bounds() for the reverse operation

with_linear_extension(linear_extension)

Return a copy of self with a different default linear extension.

EXAMPLES:

sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
sage: P.cover_relations()
[[1, 2], [1, 3], [2, 4], [2, 6], [3, 6], [4, 12], [6, 12]]
sage: list(P)
[1, 2, 3, 4, 6, 12]
sage: Q = P.with_linear_extension([1,3,2,6,4,12])
sage: list(Q)
[1, 3, 2, 6, 4, 12]
sage: Q.cover_relations()
[[1, 3], [1, 2], [3, 6], [2, 6], [2, 4], [6, 12], [4, 12]]


Note

With the current implementation, this requires relabeling the internal DiGraph which is $$O(n+m)$$, where $$n$$ is the number of elements and $$m$$ the number of cover relations.

without_bounds()

Return the poset without its top and bottom elements.

This is useful as an input for the method order_complex().

If there is either no top or no bottom element, this raises a TypeError.

EXAMPLES:

sage: P = posets.PentagonPoset()
sage: Q = P.without_bounds(); Q
Finite poset containing 3 elements
sage: Q.cover_relations()
[[2, 3]]

sage: P = posets.DiamondPoset(5)
sage: Q = P.without_bounds(); Q
Finite poset containing 3 elements
sage: Q.cover_relations()
[]


with_bounds() for the reverse operation

zeta_polynomial()

Return the zeta polynomial of the poset.

The zeta polynomial of a poset is the unique polynomial $$Z(q)$$ such that for every integer $$m > 1$$, $$Z(m)$$ is the number of weakly increasing sequences $$x_1 \leq x_2 \leq \dots \leq x_{m-1}$$ of elements of the poset.

The polynomial $$Z(q)$$ is integral-valued, but generally does not have integer coefficients. It can be computed as

$Z(q) = \sum_{k \geq 1} \dbinom{q-2}{k-1} c_k,$

where $$c_k$$ is the number of all chains of length $$k$$ in the poset.

In particular, $$Z(2)$$ is the number of vertices and $$Z(3)$$ is the number of intervals.

EXAMPLES:

sage: posets.ChainPoset(2).zeta_polynomial()
q
sage: posets.ChainPoset(3).zeta_polynomial()
1/2*q^2 + 1/2*q

sage: P = posets.PentagonPoset()
sage: P.zeta_polynomial()
1/6*q^3 + q^2 - 1/6*q

sage: P = posets.DiamondPoset(5)
sage: P.zeta_polynomial()
3/2*q^2 - 1/2*q

class sage.combinat.posets.posets.FinitePosets_n(n)

The finite enumerated set of all posets on $$n$$ elements, up to an isomorphism.

EXAMPLES:

sage: P = Posets(3)
sage: P.cardinality()
5
sage: for p in P: print(p.cover_relations())
[]
[[1, 2]]
[[0, 1], [0, 2]]
[[0, 1], [1, 2]]
[[1, 2], [0, 2]]

cardinality(from_iterator=False)

Return the cardinality of this object.

Note

By default, this returns pre-computed values obtained from the On-Line Encyclopedia of Integer Sequences (OEIS sequence A000112). To override this, pass the argument from_iterator=True.

EXAMPLES:

sage: P = Posets(3)
sage: P.cardinality()
5
sage: P.cardinality(from_iterator=True)
5

sage.combinat.posets.posets.Poset(data=None, element_labels=None, cover_relations=False, linear_extension=False, category=None, facade=None, key=None)

Construct a finite poset from various forms of input data.

INPUT:

• data – different input are accepted by this constructor:

1. A two-element list or tuple (E, R), where E is a collection of elements of the poset and R is a collection of relations x <= y, each represented as a two-element list/tuple/iterable such as [x, y]. The poset is then the transitive closure of the provided relations. If cover_relations=True, then R is assumed to contain exactly the cover relations of the poset. If E is empty, then E is taken to be the set of elements appearing in the relations R.

2. A two-element list or tuple (E, f), where E is the set of elements of the poset and f is a function such that, for any pair x, y of elements of E, f(x, y) returns whether x <= y. If cover_relations=True, then f(x, y) should instead return whether x is covered by y.

3. A dictionary of upper covers: data[x] is a list of the elements that cover the element $$x$$ in the poset.

4. A list or tuple of upper covers: data[x] is a list of the elements that cover the element $$x$$ in the poset.

If the set of elements is not a set of consecutive integers starting from zero, then:

• every element must appear in the data, for example in its own entry.
• data must be ordered in the same way as sorted elements.

Warning

If data is a list or tuple of length $$2$$, then it is handled by the case 2 above.

5. An acyclic, loop-free and multi-edge free DiGraph. If cover_relations is True, then the edges of the digraph are assumed to correspond to the cover relations of the poset. Otherwise, the cover relations are computed.

6. A previously constructed poset (the poset itself is returned).

• element_labels – (default: None); an optional list or dictionary of objects that label the poset elements.

• cover_relations – a boolean (default: False); whether the data can be assumed to describe a directed acyclic graph whose arrows are cover relations; otherwise, the cover relations are first computed.

• linear_extension – a boolean (default: False); whether to use the provided list of elements as default linear extension for the poset; otherwise a linear extension is computed. If the data is given as the pair (E, f), then E is taken to be the linear extension.

• facade – a boolean or None (default); whether the Poset()’s elements should be wrapped to make them aware of the Poset they belong to.

• If facade = True, the Poset()’s elements are exactly those given as input.
• If facade = False, the Poset()’s elements will become PosetElement objects.
• If facade = None (default) the expected behaviour is the behaviour of facade = True, unless the opposite can be deduced from the context (i.e. for instance if a Poset() is built from another Poset(), itself built with facade = False)

OUTPUT:

FinitePoset – an instance of the FinitePoset class.

If category is specified, then the poset is created in this category instead of FinitePosets.

EXAMPLES:

1. Elements and cover relations:

sage: elms = [1,2,3,4,5,6,7]
sage: rels = [[1,2],[3,4],[4,5],[2,5]]
sage: Poset((elms, rels), cover_relations = True, facade = False)
Finite poset containing 7 elements


Elements and non-cover relations:

sage: elms = [1,2,3,4]
sage: rels = [[1,2],[1,3],[1,4],[2,3],[2,4],[3,4]]
sage: P = Poset( [elms,rels] ,cover_relations=False); P
Finite poset containing 4 elements
sage: P.cover_relations()
[[1, 2], [2, 3], [3, 4]]

2. Elements and function: the standard permutations of [1, 2, 3, 4] with the Bruhat order:

sage: elms = Permutations(4)
sage: fcn = lambda p,q : p.bruhat_lequal(q)
sage: Poset((elms, fcn))
Finite poset containing 24 elements


With a function that identifies the cover relations: the set partitions of $$\{1, 2, 3\}$$ ordered by refinement:

sage: elms = SetPartitions(3)
sage: def fcn(A, B):
....:     if len(A) != len(B)+1:
....:         return False
....:     for a in A:
....:         if not any(set(a).issubset(b) for b in B):
....:             return False
....:     return True
sage: Poset((elms, fcn), cover_relations=True)
Finite poset containing 5 elements

3. A dictionary of upper covers:

sage: Poset({'a':['b','c'], 'b':['d'], 'c':['d'], 'd':[]})
Finite poset containing 4 elements

4. A list of upper covers, with range(5) as set of vertices:

sage: Poset([[1,2],,,,[]])
Finite poset containing 5 elements


A list of upper covers, with letters as vertices:

sage: Poset([["a","b"],["b","c"],["c"]])
Finite poset containing 3 elements


A list of upper covers and a dictionary of labels:

sage: elm_labs = {0:"a",1:"b",2:"c",3:"d",4:"e"}
sage: P = Poset([[1,2],,,,[]], elm_labs, facade=False)
sage: P.list()
[a, b, c, d, e]


Warning

The special case where the argument data is a list or tuple of length 2 is handled by the case 2. So you cannot use this method to input a 2-element poset.

5. An acyclic DiGraph.

sage: dag = DiGraph({0:[2,3], 1:[3,4], 2:, 3:, 4:})
sage: Poset(dag)
Finite poset containing 6 elements


Any directed acyclic graph without loops or multiple edges, as long as cover_relations=False:

sage: dig = DiGraph({0:[2,3], 1:[3,4,5], 2:, 3:, 4:})
sage: dig.allows_multiple_edges()
False
sage: dig.allows_loops()
False
sage: dig.transitive_reduction() == dig
False
sage: Poset(dig, cover_relations=False)
Finite poset containing 6 elements
sage: Poset(dig, cover_relations=True)
Traceback (most recent call last):
...
ValueError: Hasse diagram is not transitively reduced


Default Linear extension

Every poset $$P$$ obtained with Poset comes equipped with a default linear extension, which is also used for enumerating its elements. By default, this linear extension is computed, and has no particular significance:

sage: P = Poset((divisors(12), attrcall("divides")))
sage: P.list()
[1, 2, 4, 3, 6, 12]
sage: P.linear_extension()
[1, 2, 4, 3, 6, 12]


You may enforce a specific linear extension using the linear_extension option:

sage: P = Poset((divisors(12), attrcall("divides")), linear_extension=True)
sage: P.list()
[1, 2, 3, 4, 6, 12]
sage: P.linear_extension()
[1, 2, 3, 4, 6, 12]


Depending on popular request, Poset might eventually get modified to always use the provided list of elements as default linear extension, when it is one.

When facade = False, the elements of a poset are wrapped so as to make them aware that they belong to that poset:

sage: P = Poset(DiGraph({'d':['c','b'],'c':['a'],'b':['a']}), facade = False)
sage: d,c,b,a = list(P)
sage: a.parent() is P
True


This allows for comparing elements according to $$P$$:

sage: c < a
True


However, this may have surprising effects:

sage: my_elements = ['a','b','c','d']
sage: any(x in my_elements for x in P)
False


and can be annoying when one wants to manipulate the elements of the poset:

sage: a + b
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for +: 'Finite poset containing 4 elements' and 'Finite poset containing 4 elements'
sage: a.element + b.element
'ab'


sage: P = Poset(DiGraph({'d':['c','b'],'c':['a'],'b':['a']}))


In this example, the elements of the poset remain plain strings:

sage: d,c,b,a = list(P)
sage: type(a)
<... 'str'>


Of course, those strings are not aware of $$P$$. So to compare two such strings, one needs to query $$P$$:

sage: a < b
True
sage: P.lt(a,b)
False


which models the usual mathematical notation $$a <_P b$$.

Most operations seem to still work, but at this point there is no guarantee whatsoever:

sage: P.list()
['d', 'c', 'b', 'a']
sage: P.principal_order_ideal('a')
['d', 'c', 'b', 'a']
sage: P.principal_order_ideal('b')
['d', 'b']
sage: P.principal_order_ideal('d')
['d']
sage: TestSuite(P).run()


Warning

DiGraph is used to construct the poset, and the vertices of a DiGraph are converted to plain Python int’s if they are Integer’s:

sage: G = DiGraph({0:[2,3], 1:[3,4], 2:, 3:, 4:})
sage: type(G.vertices())
<... 'int'>


This is worked around by systematically converting back the vertices of a poset to Integer’s if they are int’s:

sage: P = Poset((divisors(15), attrcall("divides")), facade = False)
sage: type(P.an_element().element)
<... 'sage.rings.integer.Integer'>

sage: P = Poset((divisors(15), attrcall("divides")), facade=True)
sage: type(P.an_element())
<... 'sage.rings.integer.Integer'>


This may be abusive:

sage: P = Poset((range(5), operator.le), facade = True)
sage: P.an_element().parent()
Integer Ring


Unique representation

As most parents, Poset have unique representation (see UniqueRepresentation). Namely if two posets are created from two equal data, then they are not only equal but actually identical:

sage: data1 = [[1,2],,]
sage: data2 = [[1,2],,]
sage: P1 = Poset(data1)
sage: P2 = Poset(data2)
sage: P1 == P2
True
sage: P1 is P2
True


In situations where this behaviour is not desired, one can use the key option:

sage: P1 = Poset(data1, key = "foo")
sage: P2 = Poset(data2, key = "bar")
sage: P1 is P2
False
sage: P1 == P2
False


key can be any hashable value and is passed down to UniqueRepresentation. It is otherwise ignored by the poset constructor.

sage.combinat.posets.posets.is_poset(dig)

Return True if a directed graph is acyclic and transitively reduced, and False otherwise.

EXAMPLES:

sage: from sage.combinat.posets.posets import is_poset
sage: dig = DiGraph({0:[2, 3], 1:[3, 4, 5], 2:, 3:, 4:})
sage: is_poset(dig)
False
sage: is_poset(dig.transitive_reduction())
True