Composition Tableaux#
AUTHORS:
Chris Berg, Jeff Ferreira (2012-9): Initial version
- class sage.combinat.composition_tableau.CompositionTableau(parent, t)#
Bases:
CombinatorialElement
A composition tableau.
A composition tableau \(t\) of shape \(I = (I_1, \ldots, I_{\ell})\) is an array of boxes in rows, \(I_i\) boxes in row \(i\), filled with positive integers such that:
the entries in the rows of \(t\) weakly decrease from left to right,
the left-most column of \(t\) strictly increase from top to bottom.
Add zero entries to the rows of \(t\) until the resulting array is rectangular of shape \(\ell \times m\). For \(1 \leq i < j \leq \ell, 2 \leq k \leq m\) and \((t(j,k) \neq 0\), and also if \(t(j,k) \geq t(i,k))\) implies \(t(j,k) > t(i,k-1).\)
INPUT:
t
– A list of lists
EXAMPLES:
sage: CompositionTableau([[1],[2,2]]) [[1], [2, 2]] sage: CompositionTableau([[1],[3,2],[4,4]]) [[1], [3, 2], [4, 4]] sage: CompositionTableau([]) []
- descent_composition()#
Return the composition corresponding to the set of all \(i\) that do not have \(i+1\) appearing strictly to the left of \(i\) in
self
.EXAMPLES:
sage: CompositionTableau([[1],[3,2],[4,4]]).descent_composition() [1, 2, 2]
- descent_set()#
Return the set of all \(i\) that do not have \(i+1\) appearing strictly to the left of \(i\) in
self
.EXAMPLES:
sage: CompositionTableau([[1],[3,2],[4,4]]).descent_set() [1, 3]
- is_standard()#
Return
True
ifself
is a standard composition tableau andFalse
otherwise.EXAMPLES:
sage: CompositionTableau([[1,1],[3,2],[4,4,3]]).is_standard() False sage: CompositionTableau([[2,1],[3],[4]]).is_standard() True
- pp()#
Return a pretty print string of
self
.EXAMPLES:
sage: CompositionTableau([[1],[3,2],[4,4]]).pp() 1 3 2 4 4
- shape_composition()#
Return a Composition object which is the shape of
self
.EXAMPLES:
sage: CompositionTableau([[1,1],[3,2],[4,4,3]]).shape_composition() [2, 2, 3] sage: CompositionTableau([[2,1],[3],[4]]).shape_composition() [2, 1, 1]
- shape_partition()#
Return a partition which is the shape of
self
sorted into weakly decreasing order.EXAMPLES:
sage: CompositionTableau([[1,1],[3,2],[4,4,3]]).shape_partition() [3, 2, 2] sage: CompositionTableau([[2,1],[3],[4]]).shape_partition() [2, 1, 1]
- size()#
Return the number of boxes in
self
.EXAMPLES:
sage: CompositionTableau([[1],[3,2],[4,4]]).size() 5
- weight()#
Return a composition where entry \(i\) is the number of times that \(i\) appears in
self
.EXAMPLES:
sage: CompositionTableau([[1],[3,2],[4,4]]).weight() [1, 1, 1, 2, 0]
- class sage.combinat.composition_tableau.CompositionTableaux(**kwds)#
Bases:
UniqueRepresentation
,Parent
Composition tableaux.
INPUT:
Keyword arguments:
size
– the size of the composition tableauxshape
– the shape of the composition tableauxmax_entry
– the maximum entry for the composition tableaux
Positional arguments:
The first argument is interpreted as
size
orshape
depending on whether it is an integer or a composition.
EXAMPLES:
sage: CT = CompositionTableaux(3); CT Composition Tableaux of size 3 and maximum entry 3 sage: list(CT) [[[1], [2], [3]], [[1], [2, 2]], [[1], [3, 2]], [[1], [3, 3]], [[2], [3, 3]], [[1, 1], [2]], [[1, 1], [3]], [[2, 1], [3]], [[2, 2], [3]], [[1, 1, 1]], [[2, 1, 1]], [[2, 2, 1]], [[2, 2, 2]], [[3, 1, 1]], [[3, 2, 1]], [[3, 2, 2]], [[3, 3, 1]], [[3, 3, 2]], [[3, 3, 3]]] sage: CT = CompositionTableaux([1,2,1]); CT Composition tableaux of shape [1, 2, 1] and maximum entry 4 sage: list(CT) [[[1], [2, 2], [3]], [[1], [2, 2], [4]], [[1], [3, 2], [4]], [[1], [3, 3], [4]], [[2], [3, 3], [4]]] sage: CT = CompositionTableaux(shape=[1,2,1],max_entry=3); CT Composition tableaux of shape [1, 2, 1] and maximum entry 3 sage: list(CT) [[[1], [2, 2], [3]]] sage: CT = CompositionTableaux(2,max_entry=3); CT Composition Tableaux of size 2 and maximum entry 3 sage: list(CT) [[[1], [2]], [[1], [3]], [[2], [3]], [[1, 1]], [[2, 1]], [[2, 2]], [[3, 1]], [[3, 2]], [[3, 3]]] sage: CT = CompositionTableaux(0); CT Composition Tableaux of size 0 and maximum entry 0 sage: list(CT) [[]]
- Element#
alias of
CompositionTableau
- class sage.combinat.composition_tableau.CompositionTableauxBacktracker(shape, max_entry=None)#
Bases:
GenericBacktracker
A backtracker class for generating sets of composition tableaux.
- get_next_pos(ii, jj)#
EXAMPLES:
sage: from sage.combinat.composition_tableau import CompositionTableauxBacktracker sage: T = CompositionTableau([[2,1],[5,4,3,2],[6],[7,7,6]]) sage: n = CompositionTableauxBacktracker(T.shape_composition()) sage: n.get_next_pos(1,1) (1, 2)
- class sage.combinat.composition_tableau.CompositionTableaux_all(max_entry=None)#
Bases:
CompositionTableaux
,DisjointUnionEnumeratedSets
All composition tableaux.
- an_element()#
Return a particular element of
self
.EXAMPLES:
sage: CT = CompositionTableaux() sage: CT.an_element() [[1, 1], [2]]
- class sage.combinat.composition_tableau.CompositionTableaux_shape(comp, max_entry=None)#
Bases:
CompositionTableaux
Composition tableaux of a fixed shape
comp
with a given max entry.INPUT:
comp
– a composition.max_entry
– a nonnegative integer. This keyword argument defaults to the size ofcomp
.
- an_element()#
Return a particular element of
CompositionTableaux_shape
.EXAMPLES:
sage: CT = CompositionTableaux([1,2,1]) sage: CT.an_element() [[1], [2, 2], [3]]
- class sage.combinat.composition_tableau.CompositionTableaux_size(n, max_entry=None)#
Bases:
CompositionTableaux
Composition tableaux of a fixed size \(n\).
INPUT:
n
– a nonnegative integer.max_entry
– a nonnegative integer. This keyword argument defaults ton
.
OUTPUT:
The class of composition tableaux of size
n
.