Integer compositions¶
A composition \(c\) of a nonnegative integer \(n\) is a list of positive integers (the parts of the composition) with total sum \(n\).
This module provides tools for manipulating compositions and enumerated sets of compositions.
EXAMPLES:
sage: Composition([5, 3, 1, 3])
[5, 3, 1, 3]
sage: list(Compositions(4))
[[1, 1, 1, 1], [1, 1, 2], [1, 2, 1], [1, 3], [2, 1, 1], [2, 2], [3, 1], [4]]
AUTHORS:
Mike Hansen, Nicolas M. Thiery
MuPAD-Combinat developers (algorithms and design inspiration)
Travis Scrimshaw (2013-02-03): Removed
CombinatorialClass
- class sage.combinat.composition.Composition(parent, *args, **kwds)¶
Bases:
sage.combinat.combinat.CombinatorialElement
Integer compositions
A composition of a nonnegative integer \(n\) is a list \((i_1, \ldots, i_k)\) of positive integers with total sum \(n\).
EXAMPLES:
The simplest way to create a composition is by specifying its entries as a list, tuple (or other iterable):
sage: Composition([3,1,2]) [3, 1, 2] sage: Composition((3,1,2)) [3, 1, 2] sage: Composition(i for i in range(2,5)) [2, 3, 4]
You can also create a composition from its code. The code of a composition \((i_1, i_2, \ldots, i_k)\) of \(n\) is a list of length \(n\) that consists of a \(1\) followed by \(i_1-1\) zeros, then a \(1\) followed by \(i_2-1\) zeros, and so on.
sage: Composition([4,1,2,3,5]).to_code() [1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0] sage: Composition(code=_) [4, 1, 2, 3, 5] sage: Composition([3,1,2,3,5]).to_code() [1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0] sage: Composition(code=_) [3, 1, 2, 3, 5]
You can also create the composition of \(n\) corresponding to a subset of \(\{1, 2, \ldots, n-1\}\) under the bijection that maps the composition \((i_1, i_2, \ldots, i_k)\) of \(n\) to the subset \(\{i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + \cdots + i_{k-1}\}\) (see
to_subset()
):sage: Composition(from_subset=({1, 2, 4}, 5)) [1, 1, 2, 1] sage: Composition([1, 1, 2, 1]).to_subset() {1, 2, 4}
The following notation equivalently specifies the composition from the set \(\{i_1 - 1, i_1 + i_2 - 1, i_1 + i_2 + i_3 - 1, \dots, i_1 + \cdots + i_{k-1} - 1, n-1\}\) or \(\{i_1 - 1, i_1 + i_2 - 1, i_1 + i_2 + i_3 - 1, \dots, i_1 + \cdots + i_{k-1} - 1\}\) and \(n\). This provides compatibility with Python’s \(0\)-indexing.
sage: Composition(descents=[1,0,4,8,11]) [1, 1, 3, 4, 3] sage: Composition(descents=[0,1,3,4]) [1, 1, 2, 1] sage: Composition(descents=([0,1,3],5)) [1, 1, 2, 1] sage: Composition(descents=({0,1,3},5)) [1, 1, 2, 1]
EXAMPLES:
sage: C = Composition([3,1,2]) sage: TestSuite(C).run()
- complement()¶
Return the complement of the composition
self
.The complement of a composition \(I\) is defined as follows:
If \(I\) is the empty composition, then the complement is the empty composition as well. Otherwise, let \(S\) be the descent set of \(I\) (that is, the subset \(\{ i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}\) of \(\{ 1, 2, \ldots, |I|-1 \}\), where \(I\) is written as \((i_1, i_2, \ldots, i_k)\)). Then, the complement of \(I\) is defined as the composition of size \(|I|\) whose descent set is \(\{ 1, 2, \ldots, |I|-1 \} \setminus S\).
The complement of a composition \(I\) also is the reverse composition (
reversed()
) of the conjugate (conjugate()
) of \(I\).EXAMPLES:
sage: Composition([1, 1, 3, 1, 2, 1, 3]).conjugate() [1, 1, 3, 3, 1, 3] sage: Composition([1, 1, 3, 1, 2, 1, 3]).complement() [3, 1, 3, 3, 1, 1]
- conjugate()¶
Return the conjugate of the composition
self
.The conjugate of a composition \(I\) is defined as the complement (see
complement()
) of the reverse composition (seereversed()
) of \(I\).An equivalent definition of the conjugate goes by saying that the ribbon shape of the conjugate of a composition \(I\) is the conjugate of the ribbon shape of \(I\). (The ribbon shape of a composition is returned by
to_skew_partition()
.)This implementation uses the algorithm from mupad-combinat.
EXAMPLES:
sage: Composition([1, 1, 3, 1, 2, 1, 3]).conjugate() [1, 1, 3, 3, 1, 3]
The ribbon shape of the conjugate of \(I\) is the conjugate of the ribbon shape of \(I\):
sage: all( I.conjugate().to_skew_partition() ....: == I.to_skew_partition().conjugate() ....: for I in Compositions(4) ) True
- descents(final_descent=False)¶
This gives one fewer than the partial sums of the composition.
This is here to maintain some sort of backward compatibility, even through the original implementation was broken (it gave the wrong answer). The same information can be found in
partial_sums()
.See also
INPUT:
final_descent
– (Default:False
) a boolean integer
OUTPUT:
the list of partial sums of
self
with each part decremented by \(1\). This includes the sum of all entries whenfinal_descent
isTrue
.
EXAMPLES:
sage: c = Composition([2,1,3,2]) sage: c.descents() [1, 2, 5] sage: c.descents(final_descent=True) [1, 2, 5, 7]
- fatten(grouping)¶
Return the composition fatter than
self
, obtained by grouping together consecutive parts according togrouping
.INPUT:
grouping
– a composition whose sum is the length ofself
EXAMPLES:
Let us start with the composition:
sage: c = Composition([4,5,2,7,1])
With
grouping
equal to \((1, \ldots, 1)\), \(c\) is left unchanged:sage: c.fatten(Composition([1,1,1,1,1])) [4, 5, 2, 7, 1]
With
grouping
equal to \((\ell)\) where \(\ell\) is the length of \(c\), this yields the coarsest composition above \(c\):sage: c.fatten(Composition([5])) [19]
Other values for
grouping
yield (all the) other compositions coarser than \(c\):sage: c.fatten(Composition([2,1,2])) [9, 2, 8] sage: c.fatten(Composition([3,1,1])) [11, 7, 1]
- fatter()¶
Return the set of compositions which are fatter than
self
.Complexity for generation: \(O(|c|)\) memory, \(O(|r|)\) time where \(|c|\) is the size of
self
and \(r\) is the result.EXAMPLES:
sage: C = Composition([4,5,2]).fatter() sage: C.cardinality() 4 sage: list(C) [[4, 5, 2], [4, 7], [9, 2], [11]]
Some extreme cases:
sage: list(Composition([5]).fatter()) [[5]] sage: list(Composition([]).fatter()) [[]] sage: list(Composition([1,1,1,1]).fatter()) == list(Compositions(4)) True
- finer()¶
Return the set of compositions which are finer than
self
.EXAMPLES:
sage: C = Composition([3,2]).finer() sage: C.cardinality() 8 sage: C.list() [[1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 2, 1, 1], [1, 2, 2], [2, 1, 1, 1], [2, 1, 2], [3, 1, 1], [3, 2]] sage: Composition([]).finer() {[]}
- inf(other, check=True)¶
Return the meet of
self
with a compositionother
of the same size.The meet of two compositions \(I\) and \(J\) of size \(n\) is the finest composition of \(n\) which is coarser than each of \(I\) and \(J\). It can be described as the composition whose descent set is the intersection of the descent sets of \(I\) and \(J\).
INPUT:
other
– composition of same size asself
check
– (default:True
) a Boolean determining whether to check the input compositions for having the same size
OUTPUT:
the meet of the compositions
self
andother
EXAMPLES:
sage: Composition([3, 1, 1, 3, 1]).meet([4, 3, 2]) [4, 5] sage: Composition([9, 6]).meet([1, 3, 6, 3, 2]) [15] sage: Composition([9, 6]).meet([1, 3, 5, 1, 3, 2]) [9, 6] sage: Composition([1, 1, 1, 1, 1]).meet([3, 2]) [3, 2] sage: Composition([4, 2]).meet([3, 3]) [6] sage: Composition([]).meet([]) [] sage: Composition([1]).meet([1]) [1]
Let us verify on small examples that the meet of \(I\) and \(J\) is coarser than both of \(I\) and \(J\):
sage: all( all( I.is_finer(I.meet(J)) and ....: J.is_finer(I.meet(J)) ....: for J in Compositions(4) ) ....: for I in Compositions(4) ) True
and is the finest composition to do so:
sage: all( all( all( I.meet(J).is_finer(K) ....: for K in I.fatter() ....: if J.is_finer(K) ) ....: for J in Compositions(3) ) ....: for I in Compositions(3) ) True
The descent set of the meet of \(I\) and \(J\) is the intersection of the descent sets of \(I\) and \(J\):
sage: def test_meet(n): ....: return all( all( I.to_subset().intersection(J.to_subset()) ....: == I.meet(J).to_subset() ....: for J in Compositions(n) ) ....: for I in Compositions(n) ) sage: all( test_meet(n) for n in range(1, 5) ) True
See also
AUTHORS:
Darij Grinberg (2013-09-05)
- is_finer(co2)¶
Return
True
if the compositionself
is finer than the compositionco2
; otherwise, returnFalse
.EXAMPLES:
sage: Composition([4,1,2]).is_finer([3,1,3]) False sage: Composition([3,1,3]).is_finer([4,1,2]) False sage: Composition([1,2,2,1,1,2]).is_finer([5,1,3]) True sage: Composition([2,2,2]).is_finer([4,2]) True
- join(other, check=True)¶
Return the join of
self
with a compositionother
of the same size.The join of two compositions \(I\) and \(J\) of size \(n\) is the coarsest composition of \(n\) which refines each of \(I\) and \(J\). It can be described as the composition whose descent set is the union of the descent sets of \(I\) and \(J\). It is also the concatenation of \(I_1, I_2, \cdots , I_m\), where \(I = I_1 \bullet I_2 \bullet \ldots \bullet I_m\) is the ribbon decomposition of \(I\) with respect to \(J\) (see
ribbon_decomposition()
).INPUT:
other
– composition of same size asself
check
– (default:True
) a Boolean determining whether to check the input compositions for having the same size
OUTPUT:
the join of the compositions
self
andother
EXAMPLES:
sage: Composition([3, 1, 1, 3, 1]).join([4, 3, 2]) [3, 1, 1, 2, 1, 1] sage: Composition([9, 6]).join([1, 3, 6, 3, 2]) [1, 3, 5, 1, 3, 2] sage: Composition([9, 6]).join([1, 3, 5, 1, 3, 2]) [1, 3, 5, 1, 3, 2] sage: Composition([1, 1, 1, 1, 1]).join([3, 2]) [1, 1, 1, 1, 1] sage: Composition([4, 2]).join([3, 3]) [3, 1, 2] sage: Composition([]).join([]) []
Let us verify on small examples that the join of \(I\) and \(J\) refines both of \(I\) and \(J\):
sage: all( all( I.join(J).is_finer(I) and ....: I.join(J).is_finer(J) ....: for J in Compositions(4) ) ....: for I in Compositions(4) ) True
and is the coarsest composition to do so:
sage: all( all( all( K.is_finer(I.join(J)) ....: for K in I.finer() ....: if K.is_finer(J) ) ....: for J in Compositions(3) ) ....: for I in Compositions(3) ) True
Let us check that the join of \(I\) and \(J\) is indeed the concatenation of \(I_1, I_2, \cdots , I_m\), where \(I = I_1 \bullet I_2 \bullet \ldots \bullet I_m\) is the ribbon decomposition of \(I\) with respect to \(J\):
sage: all( all( Composition.sum(I.ribbon_decomposition(J)[0]) ....: == I.join(J) for J in Compositions(4) ) ....: for I in Compositions(4) ) True
Also, the descent set of the join of \(I\) and \(J\) is the union of the descent sets of \(I\) and \(J\):
sage: all( all( I.to_subset().union(J.to_subset()) ....: == I.join(J).to_subset() ....: for J in Compositions(4) ) ....: for I in Compositions(4) ) True
See also
AUTHORS:
Darij Grinberg (2013-09-05)
- major_index()¶
Return the major index of
self
. The major index is defined as the sum of the descents.EXAMPLES:
sage: Composition([1, 1, 3, 1, 2, 1, 3]).major_index() 31
- meet(other, check=True)¶
Return the meet of
self
with a compositionother
of the same size.The meet of two compositions \(I\) and \(J\) of size \(n\) is the finest composition of \(n\) which is coarser than each of \(I\) and \(J\). It can be described as the composition whose descent set is the intersection of the descent sets of \(I\) and \(J\).
INPUT:
other
– composition of same size asself
check
– (default:True
) a Boolean determining whether to check the input compositions for having the same size
OUTPUT:
the meet of the compositions
self
andother
EXAMPLES:
sage: Composition([3, 1, 1, 3, 1]).meet([4, 3, 2]) [4, 5] sage: Composition([9, 6]).meet([1, 3, 6, 3, 2]) [15] sage: Composition([9, 6]).meet([1, 3, 5, 1, 3, 2]) [9, 6] sage: Composition([1, 1, 1, 1, 1]).meet([3, 2]) [3, 2] sage: Composition([4, 2]).meet([3, 3]) [6] sage: Composition([]).meet([]) [] sage: Composition([1]).meet([1]) [1]
Let us verify on small examples that the meet of \(I\) and \(J\) is coarser than both of \(I\) and \(J\):
sage: all( all( I.is_finer(I.meet(J)) and ....: J.is_finer(I.meet(J)) ....: for J in Compositions(4) ) ....: for I in Compositions(4) ) True
and is the finest composition to do so:
sage: all( all( all( I.meet(J).is_finer(K) ....: for K in I.fatter() ....: if J.is_finer(K) ) ....: for J in Compositions(3) ) ....: for I in Compositions(3) ) True
The descent set of the meet of \(I\) and \(J\) is the intersection of the descent sets of \(I\) and \(J\):
sage: def test_meet(n): ....: return all( all( I.to_subset().intersection(J.to_subset()) ....: == I.meet(J).to_subset() ....: for J in Compositions(n) ) ....: for I in Compositions(n) ) sage: all( test_meet(n) for n in range(1, 5) ) True
See also
AUTHORS:
Darij Grinberg (2013-09-05)
- near_concatenation(other)¶
Return the near-concatenation of two nonempty compositions
self
andother
.The near-concatenation \(I \odot J\) of two nonempty compositions \(I\) and \(J\) is defined as the composition \((i_1, i_2, \ldots , i_{n-1}, i_n + j_1, j_2, j_3, \ldots , j_m)\), where \((i_1, i_2, \ldots , i_n) = I\) and \((j_1, j_2, \ldots , j_m) = J\).
This method returns
None
if one of the two input compositions is empty.EXAMPLES:
sage: Composition([1, 1, 3]).near_concatenation(Composition([4, 1, 2])) [1, 1, 7, 1, 2] sage: Composition([6]).near_concatenation(Composition([1, 5])) [7, 5] sage: Composition([1, 5]).near_concatenation(Composition([6])) [1, 11]
- partial_sums(final=True)¶
The partial sums of the sequence defined by the entries of the composition.
If \(I = (i_1, \ldots, i_m)\) is a composition, then the partial sums of the entries of the composition are \([i_1, i_1 + i_2, \ldots, i_1 + i_2 + \cdots + i_m]\).
INPUT:
final
– (default:True
) whether or not to include the final partial sum, which is always the size of the composition.
See also
EXAMPLES:
sage: Composition([1,1,3,1,2,1,3]).partial_sums() [1, 2, 5, 6, 8, 9, 12]
With
final = False
, the last partial sum is not included:sage: Composition([1,1,3,1,2,1,3]).partial_sums(final=False) [1, 2, 5, 6, 8, 9]
- peaks()¶
Return a list of the peaks of the composition
self
. The peaks of a composition are the descents which do not immediately follow another descent.EXAMPLES:
sage: Composition([1, 1, 3, 1, 2, 1, 3]).peaks() [4, 7]
- refinement_splitting(J)¶
Return the refinement splitting of
self
according toJ
.INPUT:
J
– A composition such thatself
is finer thanJ
OUTPUT:
the unique list of compositions \((I^{(p)})_{p=1, \ldots , m}\), obtained by splitting \(I\), such that \(|I^{(p)}| = J_p\) for all \(p = 1, \ldots, m\).
See also
EXAMPLES:
sage: Composition([1,2,2,1,1,2]).refinement_splitting([5,1,3]) [[1, 2, 2], [1], [1, 2]] sage: Composition([]).refinement_splitting([]) [] sage: Composition([3]).refinement_splitting([2]) Traceback (most recent call last): ... ValueError: compositions self (= [3]) and J (= [2]) must be of the same size sage: Composition([2,1]).refinement_splitting([1,2]) Traceback (most recent call last): ... ValueError: composition J (= [2, 1]) does not refine self (= [1, 2])
- refinement_splitting_lengths(J)¶
Return the lengths of the compositions in the refinement splitting of
self
according toJ
.See also
refinement_splitting()
for the definition of refinement splittingEXAMPLES:
sage: Composition([1,2,2,1,1,2]).refinement_splitting_lengths([5,1,3]) [3, 1, 2] sage: Composition([]).refinement_splitting_lengths([]) [] sage: Composition([3]).refinement_splitting_lengths([2]) Traceback (most recent call last): ... ValueError: compositions self (= [3]) and J (= [2]) must be of the same size sage: Composition([2,1]).refinement_splitting_lengths([1,2]) Traceback (most recent call last): ... ValueError: composition J (= [2, 1]) does not refine self (= [1, 2])
- reversed()¶
Return the reverse composition of
self
.The reverse composition of a composition \((i_1, i_2, \ldots, i_k)\) is defined as the composition \((i_k, i_{k-1}, \ldots, i_1)\).
EXAMPLES:
sage: Composition([1, 1, 3, 1, 2, 1, 3]).reversed() [3, 1, 2, 1, 3, 1, 1]
- ribbon_decomposition(other, check=True)¶
Return a pair describing the ribbon decomposition of a composition
self
with respect to a compositionother
of the same size.If \(I\) and \(J\) are two compositions of the same nonzero size, then the ribbon decomposition of \(I\) with respect to \(J\) is defined as follows: Write \(I\) and \(J\) as \(I = (i_1, i_2, \ldots , i_n)\) and \(J = (j_1, j_2, \ldots , j_m)\). Then, the equality \(I = I_1 \bullet I_2 \bullet \ldots \bullet I_m\) holds for a unique \(m\)-tuple \((I_1, I_2, \ldots , I_m)\) of compositions such that each \(I_k\) has size \(j_k\) and for a unique choice of \(m-1\) signs \(\bullet\) each of which is either the concatenation sign \(\cdot\) or the near-concatenation sign \(\odot\) (see
__add__()
andnear_concatenation()
for the definitions of these two signs). This \(m\)-tuple and this choice of signs together are said to form the ribbon decomposition of \(I\) with respect to \(J\). If \(I\) and \(J\) are empty, then the same definition applies, except that there are \(0\) rather than \(m-1\) signs.See Section 4.8 of [NCSF1].
INPUT:
other
– composition of same size asself
check
– (default:True
) a Boolean determining whether to check the input compositions for having the same size
OUTPUT:
a pair
(u, v)
, whereu
is a tuple of compositions (corresponding to the \(m\)-tuple \((I_1, I_2, \ldots , I_m)\) in the above definition), andv
is a tuple of \(0\) in the above definition, with a \(0\) standing for \(\cdot\) and a \(1\) standing for \(\odot\)).
EXAMPLES:
sage: Composition([3, 1, 1, 3, 1]).ribbon_decomposition([4, 3, 2]) (([3, 1], [1, 2], [1, 1]), (0, 1)) sage: Composition([9, 6]).ribbon_decomposition([1, 3, 6, 3, 2]) (([1], [3], [5, 1], [3], [2]), (1, 1, 1, 1)) sage: Composition([9, 6]).ribbon_decomposition([1, 3, 5, 1, 3, 2]) (([1], [3], [5], [1], [3], [2]), (1, 1, 0, 1, 1)) sage: Composition([1, 1, 1, 1, 1]).ribbon_decomposition([3, 2]) (([1, 1, 1], [1, 1]), (0,)) sage: Composition([4, 2]).ribbon_decomposition([6]) (([4, 2],), ()) sage: Composition([]).ribbon_decomposition([]) ((), ())
Let us check that the defining property \(I = I_1 \bullet I_2 \bullet \ldots \bullet I_m\) is satisfied:
sage: def compose_back(u, v): ....: comp = u[0] ....: r = len(v) ....: if len(u) != r + 1: ....: raise ValueError("something is wrong") ....: for i in range(r): ....: if v[i] == 0: ....: comp += u[i + 1] ....: else: ....: comp = comp.near_concatenation(u[i + 1]) ....: return comp sage: all( all( all( compose_back(*(I.ribbon_decomposition(J))) == I ....: for J in Compositions(n) ) ....: for I in Compositions(n) ) ....: for n in range(1, 5) ) True
AUTHORS:
Darij Grinberg (2013-08-29)
- shuffle_product(other, overlap=False)¶
The (overlapping) shuffles of
self
andother
.Suppose \(I = (i_1, \ldots, i_k)\) and \(J = (j_1, \ldots, j_l)\) are two compositions. A shuffle of \(I\) and \(J\) is a composition of length \(k + l\) that contains both \(I\) and \(J\) as subsequences.
More generally, an overlapping shuffle of \(I\) and \(J\) is obtained by distributing the elements of \(I\) and \(J\) (preserving the relative ordering of these elements) among the positions of an empty list; an element of \(I\) and an element of \(J\) are permitted to share the same position, in which case they are replaced by their sum. In particular, a shuffle of \(I\) and \(J\) is an overlapping shuffle of \(I\) and \(J\).
INPUT:
other
– compositionoverlap
– boolean (default:False
); ifTrue
, the overlapping shuffle product is returned.
OUTPUT:
An enumerated set (allowing for multiplicities)
EXAMPLES:
The shuffle product of \([2,2]\) and \([1,1,3]\):
sage: alph = Composition([2,2]) sage: beta = Composition([1,1,3]) sage: S = alph.shuffle_product(beta); S Shuffle product of [2, 2] and [1, 1, 3] sage: S.list() [[2, 2, 1, 1, 3], [2, 1, 2, 1, 3], [2, 1, 1, 2, 3], [2, 1, 1, 3, 2], [1, 2, 2, 1, 3], [1, 2, 1, 2, 3], [1, 2, 1, 3, 2], [1, 1, 2, 2, 3], [1, 1, 2, 3, 2], [1, 1, 3, 2, 2]]
The overlapping shuffle product of \([2,2]\) and \([1,1,3]\):
sage: alph = Composition([2,2]) sage: beta = Composition([1,1,3]) sage: O = alph.shuffle_product(beta, overlap=True); O Overlapping shuffle product of [2, 2] and [1, 1, 3] sage: O.list() [[2, 2, 1, 1, 3], [2, 1, 2, 1, 3], [2, 1, 1, 2, 3], [2, 1, 1, 3, 2], [1, 2, 2, 1, 3], [1, 2, 1, 2, 3], [1, 2, 1, 3, 2], [1, 1, 2, 2, 3], [1, 1, 2, 3, 2], [1, 1, 3, 2, 2], [3, 2, 1, 3], [2, 3, 1, 3], [3, 1, 2, 3], [2, 1, 3, 3], [3, 1, 3, 2], [2, 1, 1, 5], [1, 3, 2, 3], [1, 2, 3, 3], [1, 3, 3, 2], [1, 2, 1, 5], [1, 1, 5, 2], [1, 1, 2, 5], [3, 3, 3], [3, 1, 5], [1, 3, 5]]
Note that the shuffle product of two compositions can include the same composition more than once since a composition can be a shuffle of two compositions in several ways. For example:
sage: w1 = Composition([1]) sage: S = w1.shuffle_product(w1); S Shuffle product of [1] and [1] sage: S.list() [[1, 1], [1, 1]] sage: O = w1.shuffle_product(w1, overlap=True); O Overlapping shuffle product of [1] and [1] sage: O.list() [[1, 1], [1, 1], [2]]
- size()¶
Return the size of
self
, that is the sum of its parts.EXAMPLES:
sage: Composition([7,1,3]).size() 11
- static sum(compositions)¶
Return the concatenation of the given compositions.
INPUT:
compositions
– a list (or iterable) of compositions
EXAMPLES:
sage: Composition.sum([Composition([1, 1, 3]), Composition([4, 1, 2]), Composition([3,1])]) [1, 1, 3, 4, 1, 2, 3, 1]
Any iterable can be provided as input:
sage: Composition.sum([Composition([i,i]) for i in [4,1,3]]) [4, 4, 1, 1, 3, 3]
Empty inputs are handled gracefully:
sage: Composition.sum([]) == Composition([]) True
- sup(other, check=True)¶
Return the join of
self
with a compositionother
of the same size.The join of two compositions \(I\) and \(J\) of size \(n\) is the coarsest composition of \(n\) which refines each of \(I\) and \(J\). It can be described as the composition whose descent set is the union of the descent sets of \(I\) and \(J\). It is also the concatenation of \(I_1, I_2, \cdots , I_m\), where \(I = I_1 \bullet I_2 \bullet \ldots \bullet I_m\) is the ribbon decomposition of \(I\) with respect to \(J\) (see
ribbon_decomposition()
).INPUT:
other
– composition of same size asself
check
– (default:True
) a Boolean determining whether to check the input compositions for having the same size
OUTPUT:
the join of the compositions
self
andother
EXAMPLES:
sage: Composition([3, 1, 1, 3, 1]).join([4, 3, 2]) [3, 1, 1, 2, 1, 1] sage: Composition([9, 6]).join([1, 3, 6, 3, 2]) [1, 3, 5, 1, 3, 2] sage: Composition([9, 6]).join([1, 3, 5, 1, 3, 2]) [1, 3, 5, 1, 3, 2] sage: Composition([1, 1, 1, 1, 1]).join([3, 2]) [1, 1, 1, 1, 1] sage: Composition([4, 2]).join([3, 3]) [3, 1, 2] sage: Composition([]).join([]) []
Let us verify on small examples that the join of \(I\) and \(J\) refines both of \(I\) and \(J\):
sage: all( all( I.join(J).is_finer(I) and ....: I.join(J).is_finer(J) ....: for J in Compositions(4) ) ....: for I in Compositions(4) ) True
and is the coarsest composition to do so:
sage: all( all( all( K.is_finer(I.join(J)) ....: for K in I.finer() ....: if K.is_finer(J) ) ....: for J in Compositions(3) ) ....: for I in Compositions(3) ) True
Let us check that the join of \(I\) and \(J\) is indeed the concatenation of \(I_1, I_2, \cdots , I_m\), where \(I = I_1 \bullet I_2 \bullet \ldots \bullet I_m\) is the ribbon decomposition of \(I\) with respect to \(J\):
sage: all( all( Composition.sum(I.ribbon_decomposition(J)[0]) ....: == I.join(J) for J in Compositions(4) ) ....: for I in Compositions(4) ) True
Also, the descent set of the join of \(I\) and \(J\) is the union of the descent sets of \(I\) and \(J\):
sage: all( all( I.to_subset().union(J.to_subset()) ....: == I.join(J).to_subset() ....: for J in Compositions(4) ) ....: for I in Compositions(4) ) True
See also
AUTHORS:
Darij Grinberg (2013-09-05)
- to_code()¶
Return the code of the composition
self
. The code of a composition \(I\) is a list of length \(\mathrm{size}(I)\) of 1s and 0s such that there is a 1 wherever a new part starts. (Exceptional case: When the composition is empty, the code is[0]
.)EXAMPLES:
sage: Composition([4,1,2,3,5]).to_code() [1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0]
- to_partition()¶
Return the partition obtained by sorting
self
into decreasing order.EXAMPLES:
sage: Composition([2,1,3]).to_partition() [3, 2, 1] sage: Composition([4,2,2]).to_partition() [4, 2, 2] sage: Composition([]).to_partition() []
- to_skew_partition(overlap=1)¶
Return the skew partition obtained from
self
. This is a skew partition whose rows have the entries ofself
as their length, taken in reverse order (so the first entry ofself
is the length of the lowermost row, etc.). The parameteroverlap
indicates the number of cells on each row that are directly below cells of the previous row. When it is set to \(1\) (its default value), the result is the ribbon shape ofself
.EXAMPLES:
sage: Composition([3,4,1]).to_skew_partition() [6, 6, 3] / [5, 2] sage: Composition([3,4,1]).to_skew_partition(overlap=0) [8, 7, 3] / [7, 3] sage: Composition([]).to_skew_partition() [] / [] sage: Composition([1,2]).to_skew_partition() [2, 1] / [] sage: Composition([2,1]).to_skew_partition() [2, 2] / [1]
- to_subset(final=False)¶
The subset corresponding to
self
under the bijection (see below) between compositions of \(n\) and subsets of \(\{1, 2, \ldots, n-1\}\).The bijection maps a composition \((i_1, \ldots, i_k)\) of \(n\) to \(\{i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + \cdots + i_{k-1}\}\).
INPUT:
final
– (default:False
) whether or not to include the final partial sum, which is always the size of the composition.
See also
EXAMPLES:
sage: Composition([1,1,3,1,2,1,3]).to_subset() {1, 2, 5, 6, 8, 9} sage: for I in Compositions(3): print(I.to_subset()) {1, 2} {1} {2} {}
With
final=True
, the sum of all the elements of the composition is included in the subset:sage: Composition([1,1,3,1,2,1,3]).to_subset(final=True) {1, 2, 5, 6, 8, 9, 12}
- wll_gt(co2)¶
Return
True
if the compositionself
is greater than the compositionco2
with respect to the wll-ordering; otherwise, returnFalse
.The wll-ordering is a total order on the set of all compositions defined as follows: A composition \(I\) is greater than a composition \(J\) if and only if one of the following conditions holds:
The size of \(I\) is greater than the size of \(J\).
The size of \(I\) equals the size of \(J\), but the length of \(I\) is greater than the length of \(J\).
The size of \(I\) equals the size of \(J\), and the length of \(I\) equals the length of \(J\), but \(I\) is lexicographically greater than \(J\).
(“wll-ordering” is short for “weight, length, lexicographic ordering”.)
EXAMPLES:
sage: Composition([4,1,2]).wll_gt([3,1,3]) True sage: Composition([7]).wll_gt([4,1,2]) False sage: Composition([8]).wll_gt([4,1,2]) True sage: Composition([3,2,2,2]).wll_gt([5,2]) True sage: Composition([]).wll_gt([3]) False sage: Composition([2,1]).wll_gt([2,1]) False sage: Composition([2,2,2]).wll_gt([4,2]) True sage: Composition([4,2]).wll_gt([2,2,2]) False sage: Composition([1,1,2]).wll_gt([2,2]) True sage: Composition([2,2]).wll_gt([1,3]) True sage: Composition([2,1,2]).wll_gt([]) True
- class sage.combinat.composition.Compositions(is_infinite=False)¶
Bases:
sage.structure.unique_representation.UniqueRepresentation
,sage.structure.parent.Parent
Set of integer compositions.
A composition \(c\) of a nonnegative integer \(n\) is a list of positive integers with total sum \(n\).
See also
EXAMPLES:
There are 8 compositions of 4:
sage: Compositions(4).cardinality() 8
Here is the list of them:
sage: Compositions(4).list() [[1, 1, 1, 1], [1, 1, 2], [1, 2, 1], [1, 3], [2, 1, 1], [2, 2], [3, 1], [4]]
You can use the
.first()
method to get the ‘first’ composition of a number:sage: Compositions(4).first() [1, 1, 1, 1]
You can also calculate the ‘next’ composition given the current one:
sage: Compositions(4).next([1,1,2]) [1, 2, 1]
If \(n\) is not specified, this returns the combinatorial class of all (non-negative) integer compositions:
sage: Compositions() Compositions of non-negative integers sage: [] in Compositions() True sage: [2,3,1] in Compositions() True sage: [-2,3,1] in Compositions() False
If \(n\) is specified, it returns the class of compositions of \(n\):
sage: Compositions(3) Compositions of 3 sage: list(Compositions(3)) [[1, 1, 1], [1, 2], [2, 1], [3]] sage: Compositions(3).cardinality() 4
The following examples show how to test whether or not an object is a composition:
sage: [3,4] in Compositions() True sage: [3,4] in Compositions(7) True sage: [3,4] in Compositions(5) False
Similarly, one can check whether or not an object is a composition which satisfies further constraints:
sage: [4,2] in Compositions(6, inner=[2,2]) True sage: [4,2] in Compositions(6, inner=[2,3]) False sage: [4,1] in Compositions(5, inner=[2,1], max_slope = 0) True
An example with incompatible constraints:
sage: [4,2] in Compositions(6, inner=[2,2], min_part=3) False
The options
length
,min_length
, andmax_length
can be used to set length constraints on the compositions. For example, the compositions of 4 of length equal to, at least, and at most 2 are given by:sage: Compositions(4, length=2).list() [[3, 1], [2, 2], [1, 3]] sage: Compositions(4, min_length=2).list() [[3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]] sage: Compositions(4, max_length=2).list() [[4], [3, 1], [2, 2], [1, 3]]
Setting both
min_length
andmax_length
to the same value is equivalent to settinglength
to this value:sage: Compositions(4, min_length=2, max_length=2).list() [[3, 1], [2, 2], [1, 3]]
The options
inner
andouter
can be used to set part-by-part containment constraints. The list of compositions of 4 bounded above by[3,1,2]
is given by:sage: list(Compositions(4, outer=[3,1,2])) [[3, 1], [2, 1, 1], [1, 1, 2]]
outer
setsmax_length
to the length of its argument. Moreover, the parts ofouter
may be infinite to clear the constraint on specific parts. This is the list of compositions of 4 of length at most 3 such that the first and third parts are at most 1:sage: Compositions(4, outer=[1,oo,1]).list() [[1, 3], [1, 2, 1]]
This is the list of compositions of 4 bounded below by
[1,1,1]
:sage: Compositions(4, inner=[1,1,1]).list() [[2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]
The options
min_slope
andmax_slope
can be used to set constraints on the slope, that is the differencep[i+1]-p[i]
of two consecutive parts. The following is the list of weakly increasing compositions of 4:sage: Compositions(4, min_slope=0).list() [[4], [2, 2], [1, 3], [1, 1, 2], [1, 1, 1, 1]]
Here are the weakly decreasing ones:
sage: Compositions(4, max_slope=0).list() [[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]
The following is the list of compositions of 4 such that two consecutive parts differ by at most one:
sage: Compositions(4, min_slope=-1, max_slope=1).list() [[4], [2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]
The constraints can be combined together in all reasonable ways. This is the list of compositions of 5 of length between 2 and 4 such that the difference between consecutive parts is between -2 and 1:
sage: Compositions(5, max_slope=1, min_slope=-2, min_length=2, max_length=4).list() [[3, 2], [3, 1, 1], [2, 3], [2, 2, 1], [2, 1, 2], [2, 1, 1, 1], [1, 2, 2], [1, 2, 1, 1], [1, 1, 2, 1], [1, 1, 1, 2]]
We can do the same thing with an outer constraint:
sage: Compositions(5, max_slope=1, min_slope=-2, min_length=2, max_length=4, outer=[2,5,2]).list() [[2, 3], [2, 2, 1], [2, 1, 2], [1, 2, 2]]
However, providing incoherent constraints may yield strange results. It is up to the user to ensure that the inner and outer compositions themselves satisfy the parts and slope constraints.
Note that if you specify
min_part=0
, then the objects produced may have parts equal to zero. This violates the internal assumptions that the composition class makes. Use at your own risk, or preferably consider usingIntegerVectors
instead:sage: Compositions(2, length=3, min_part=0).list() doctest:...: RuntimeWarning: Currently, setting min_part=0 produces Composition objects which violate internal assumptions. Calling methods on these objects may produce errors or WRONG results! [[2, 0, 0], [1, 1, 0], [1, 0, 1], [0, 2, 0], [0, 1, 1], [0, 0, 2]] sage: list(IntegerVectors(2, 3)) [[2, 0, 0], [1, 1, 0], [1, 0, 1], [0, 2, 0], [0, 1, 1], [0, 0, 2]]
The generation algorithm is constant amortized time, and handled by the generic tool
IntegerListsLex
.- Element¶
alias of
Composition
- from_code(code)¶
Return the composition from its code. The code of a composition \(I\) is a list of length \(\mathrm{size}(I)\) consisting of 1s and 0s such that there is a 1 wherever a new part starts. (Exceptional case: When the composition is empty, the code is
[0]
.)EXAMPLES:
sage: Composition([4,1,2,3,5]).to_code() [1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0] sage: Compositions().from_code(_) [4, 1, 2, 3, 5] sage: Composition([3,1,2,3,5]).to_code() [1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0] sage: Compositions().from_code(_) [3, 1, 2, 3, 5]
- from_descents(descents, nps=None)¶
Return a composition from the list of descents.
INPUT:
descents
– an iterablenps
– (default:None
) an integer orNone
OUTPUT:
The composition of
nps
whose descents are listed indescents
, assuming thatnps
is notNone
(otherwise, the last element ofdescents
is removed fromdescents
, andnps
is set to be this last element plus 1).
EXAMPLES:
sage: [x-1 for x in Composition([1, 1, 3, 4, 3]).to_subset()] [0, 1, 4, 8] sage: Compositions().from_descents([1,0,4,8],12) [1, 1, 3, 4, 3] sage: Compositions().from_descents([1,0,4,8,11]) [1, 1, 3, 4, 3]
- from_subset(S, n)¶
The composition of \(n\) corresponding to the subset
S
of \(\{1, 2, \ldots, n-1\}\) under the bijection that maps the composition \((i_1, i_2, \ldots, i_k)\) of \(n\) to the subset \(\{i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + \cdots + i_{k-1}\}\) (seeComposition.to_subset()
).INPUT:
S
– an iterable, a subset of \(\{1, 2, \ldots, n-1\}\)n
– an integer
EXAMPLES:
sage: Compositions().from_subset([2,1,5,9], 12) [1, 1, 3, 4, 3] sage: Compositions().from_subset({2,1,5,9}, 12) [1, 1, 3, 4, 3] sage: Compositions().from_subset([], 12) [12] sage: Compositions().from_subset([], 0) []
- class sage.combinat.composition.Compositions_all¶
Bases:
sage.combinat.composition.Compositions
Class of all compositions.
- subset(size=None)¶
Return the set of compositions of the given size.
EXAMPLES:
sage: C = Compositions() sage: C.subset(4) Compositions of 4 sage: C.subset(size=3) Compositions of 3
- class sage.combinat.composition.Compositions_constraints(*args, **kwds)¶
- class sage.combinat.composition.Compositions_n(n)¶
Bases:
sage.combinat.composition.Compositions
Class of compositions of a fixed \(n\).
- cardinality()¶
Return the number of compositions of \(n\).
- random_element()¶
Return a random
Composition
with uniform probability.This method generates a random binary word starting with a 1 and then uses the bijection between compositions and their code.
EXAMPLES:
sage: Compositions(5).random_element() # random [2, 1, 1, 1] sage: Compositions(0).random_element() [] sage: Compositions(1).random_element() [1]
- sage.combinat.composition.composition_iterator_fast(n)¶
Iterator over compositions of
n
yielded as lists.