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]]
>>> from sage.all import *
>>> Composition([Integer(5), Integer(3), Integer(1), Integer(3)])
[5, 3, 1, 3]
>>> list(Compositions(Integer(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. Thiéry 
- MuPAD-Combinat developers (algorithms and design inspiration) 
- Travis Scrimshaw (2013-02-03): Removed - CombinatorialClass
- class sage.combinat.composition.Composition(parent, lst)[source]¶
- Bases: - 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] - >>> from sage.all import * >>> Composition([Integer(3),Integer(1),Integer(2)]) [3, 1, 2] >>> Composition((Integer(3),Integer(1),Integer(2))) [3, 1, 2] >>> Composition(i for i in range(Integer(2),Integer(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] - >>> from sage.all import * >>> Composition([Integer(4),Integer(1),Integer(2),Integer(3),Integer(5)]).to_code() [1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0] >>> Composition(code=_) [4, 1, 2, 3, 5] >>> Composition([Integer(3),Integer(1),Integer(2),Integer(3),Integer(5)]).to_code() [1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0] >>> 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} - >>> from sage.all import * >>> Composition(from_subset=({Integer(1), Integer(2), Integer(4)}, Integer(5))) [1, 1, 2, 1] >>> Composition([Integer(1), Integer(1), Integer(2), Integer(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] - >>> from sage.all import * >>> Composition(descents=[Integer(1),Integer(0),Integer(4),Integer(8),Integer(11)]) [1, 1, 3, 4, 3] >>> Composition(descents=[Integer(0),Integer(1),Integer(3),Integer(4)]) [1, 1, 2, 1] >>> Composition(descents=([Integer(0),Integer(1),Integer(3)],Integer(5))) [1, 1, 2, 1] >>> Composition(descents=({Integer(0),Integer(1),Integer(3)},Integer(5))) [1, 1, 2, 1] - An integer composition may be regarded as a sequence. Thus it is an instance of the Python abstract base class - Sequenceallows us to check if objects behave “like” sequences based on implemented methods. Note that- collections.abc.Sequenceis not the same as- sage.structure.sequence.Sequence:- sage: import collections.abc sage: C = Composition([3,2,3]) sage: isinstance(C, collections.abc.Sequence) True sage: issubclass(C.__class__, collections.abc.Sequence) True - >>> from sage.all import * >>> import collections.abc >>> C = Composition([Integer(3),Integer(2),Integer(3)]) >>> isinstance(C, collections.abc.Sequence) True >>> issubclass(C.__class__, collections.abc.Sequence) True - Typically, instances of - collections.abc.Sequencehave a- .countmethod.- Composition.countcounts the number of parts of a specified size:- sage: C.count(3) 2 - >>> from sage.all import * >>> C.count(Integer(3)) 2 - EXAMPLES: - sage: C = Composition([3,1,2]) sage: TestSuite(C).run() - >>> from sage.all import * >>> C = Composition([Integer(3),Integer(1),Integer(2)]) >>> TestSuite(C).run() - complement()[source]¶
- 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] - >>> from sage.all import * >>> Composition([Integer(1), Integer(1), Integer(3), Integer(1), Integer(2), Integer(1), Integer(3)]).conjugate() [1, 1, 3, 3, 1, 3] >>> Composition([Integer(1), Integer(1), Integer(3), Integer(1), Integer(2), Integer(1), Integer(3)]).complement() [3, 1, 3, 3, 1, 1] 
 - conjugate()[source]¶
- Return the conjugate of the composition - self.- The conjugate of a composition \(I\) is defined as the complement (see - complement()) of the reverse composition (see- reversed()) 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] - >>> from sage.all import * >>> Composition([Integer(1), Integer(1), Integer(3), Integer(1), Integer(2), Integer(1), Integer(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() # needs sage.combinat ....: == I.to_skew_partition().conjugate() ....: for I in Compositions(4) ) True - >>> from sage.all import * >>> all( I.conjugate().to_skew_partition() # needs sage.combinat ... == I.to_skew_partition().conjugate() ... for I in Compositions(Integer(4)) ) True 
 - count(n)[source]¶
- Return the number of parts of size - n.- EXAMPLES: - sage: C = Composition([3,2,3]) sage: C.count(3) 2 sage: C.count(2) 1 sage: C.count(1) 0 - >>> from sage.all import * >>> C = Composition([Integer(3),Integer(2),Integer(3)]) >>> C.count(Integer(3)) 2 >>> C.count(Integer(2)) 1 >>> C.count(Integer(1)) 0 
 - descents(final_descent=False)[source]¶
- 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– boolean (default:- False)
 - OUTPUT: - the list of partial sums of - selfwith each part decremented by \(1\). This includes the sum of all entries when- final_descentis- True.
 - EXAMPLES: - sage: c = Composition([2,1,3,2]) sage: c.descents() [1, 2, 5] sage: c.descents(final_descent=True) [1, 2, 5, 7] - >>> from sage.all import * >>> c = Composition([Integer(2),Integer(1),Integer(3),Integer(2)]) >>> c.descents() [1, 2, 5] >>> c.descents(final_descent=True) [1, 2, 5, 7] 
 - fatten(grouping)[source]¶
- Return the composition fatter than - self, obtained by grouping together consecutive parts according to- grouping.- INPUT: - grouping– a composition whose sum is the length of- self
 - EXAMPLES: - Let us start with the composition: - sage: c = Composition([4,5,2,7,1]) - >>> from sage.all import * >>> c = Composition([Integer(4),Integer(5),Integer(2),Integer(7),Integer(1)]) - With - groupingequal to \((1, \ldots, 1)\), \(c\) is left unchanged:- sage: c.fatten(Composition([1,1,1,1,1])) [4, 5, 2, 7, 1] - >>> from sage.all import * >>> c.fatten(Composition([Integer(1),Integer(1),Integer(1),Integer(1),Integer(1)])) [4, 5, 2, 7, 1] - With - groupingequal to \((\ell)\) where \(\ell\) is the length of \(c\), this yields the coarsest composition above \(c\):- sage: c.fatten(Composition([5])) [19] - >>> from sage.all import * >>> c.fatten(Composition([Integer(5)])) [19] - Other values for - groupingyield (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] - >>> from sage.all import * >>> c.fatten(Composition([Integer(2),Integer(1),Integer(2)])) [9, 2, 8] >>> c.fatten(Composition([Integer(3),Integer(1),Integer(1)])) [11, 7, 1] 
 - fatter()[source]¶
- 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 - selfand \(r\) 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]] - >>> from sage.all import * >>> C = Composition([Integer(4),Integer(5),Integer(2)]).fatter() >>> C.cardinality() 4 >>> 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 - >>> from sage.all import * >>> list(Composition([Integer(5)]).fatter()) [[5]] >>> list(Composition([]).fatter()) [[]] >>> list(Composition([Integer(1),Integer(1),Integer(1),Integer(1)]).fatter()) == list(Compositions(Integer(4))) True 
 - finer()[source]¶
- 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() {[]} - >>> from sage.all import * >>> C = Composition([Integer(3),Integer(2)]).finer() >>> C.cardinality() 8 >>> 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]] >>> Composition([]).finer() {[]} 
 - inf(other, check=True)[source]¶
- Return the meet of - selfwith a composition- otherof 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 as- self
- check– boolean (default:- True); whether to check the input compositions for having the same size
 - OUTPUT: the meet of the compositions - selfand- other- 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] - >>> from sage.all import * >>> Composition([Integer(3), Integer(1), Integer(1), Integer(3), Integer(1)]).meet([Integer(4), Integer(3), Integer(2)]) [4, 5] >>> Composition([Integer(9), Integer(6)]).meet([Integer(1), Integer(3), Integer(6), Integer(3), Integer(2)]) [15] >>> Composition([Integer(9), Integer(6)]).meet([Integer(1), Integer(3), Integer(5), Integer(1), Integer(3), Integer(2)]) [9, 6] >>> Composition([Integer(1), Integer(1), Integer(1), Integer(1), Integer(1)]).meet([Integer(3), Integer(2)]) [3, 2] >>> Composition([Integer(4), Integer(2)]).meet([Integer(3), Integer(3)]) [6] >>> Composition([]).meet([]) [] >>> Composition([Integer(1)]).meet([Integer(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 - >>> from sage.all import * >>> all( all( I.is_finer(I.meet(J)) and ... J.is_finer(I.meet(J)) ... for J in Compositions(Integer(4)) ) ... for I in Compositions(Integer(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 - >>> from sage.all import * >>> all( all( all( I.meet(J).is_finer(K) ... for K in I.fatter() ... if J.is_finer(K) ) ... for J in Compositions(Integer(3)) ) ... for I in Compositions(Integer(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 - >>> from sage.all import * >>> 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) ) >>> all( test_meet(n) for n in range(Integer(1), Integer(5)) ) True - See also - AUTHORS: - Darij Grinberg (2013-09-05) 
 
 - is_finer(co2)[source]¶
- Return - Trueif the composition- selfis finer than the composition- co2; otherwise, return- False.- 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 - >>> from sage.all import * >>> Composition([Integer(4),Integer(1),Integer(2)]).is_finer([Integer(3),Integer(1),Integer(3)]) False >>> Composition([Integer(3),Integer(1),Integer(3)]).is_finer([Integer(4),Integer(1),Integer(2)]) False >>> Composition([Integer(1),Integer(2),Integer(2),Integer(1),Integer(1),Integer(2)]).is_finer([Integer(5),Integer(1),Integer(3)]) True >>> Composition([Integer(2),Integer(2),Integer(2)]).is_finer([Integer(4),Integer(2)]) True 
 - join(other, check=True)[source]¶
- Return the join of - selfwith a composition- otherof 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 as- self
- check– boolean (default:- True); whether to check the input compositions for having the same size
 - OUTPUT: the join of the compositions - selfand- other- 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([]) [] - >>> from sage.all import * >>> Composition([Integer(3), Integer(1), Integer(1), Integer(3), Integer(1)]).join([Integer(4), Integer(3), Integer(2)]) [3, 1, 1, 2, 1, 1] >>> Composition([Integer(9), Integer(6)]).join([Integer(1), Integer(3), Integer(6), Integer(3), Integer(2)]) [1, 3, 5, 1, 3, 2] >>> Composition([Integer(9), Integer(6)]).join([Integer(1), Integer(3), Integer(5), Integer(1), Integer(3), Integer(2)]) [1, 3, 5, 1, 3, 2] >>> Composition([Integer(1), Integer(1), Integer(1), Integer(1), Integer(1)]).join([Integer(3), Integer(2)]) [1, 1, 1, 1, 1] >>> Composition([Integer(4), Integer(2)]).join([Integer(3), Integer(3)]) [3, 1, 2] >>> 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 - >>> from sage.all import * >>> all( all( I.join(J).is_finer(I) and ... I.join(J).is_finer(J) ... for J in Compositions(Integer(4)) ) ... for I in Compositions(Integer(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 - >>> from sage.all import * >>> all( all( all( K.is_finer(I.join(J)) ... for K in I.finer() ... if K.is_finer(J) ) ... for J in Compositions(Integer(3)) ) ... for I in Compositions(Integer(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 - >>> from sage.all import * >>> all( all( Composition.sum(I.ribbon_decomposition(J)[Integer(0)]) ... == I.join(J) for J in Compositions(Integer(4)) ) ... for I in Compositions(Integer(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 - >>> from sage.all import * >>> all( all( I.to_subset().union(J.to_subset()) ... == I.join(J).to_subset() ... for J in Compositions(Integer(4)) ) ... for I in Compositions(Integer(4)) ) True - See also - AUTHORS: - Darij Grinberg (2013-09-05) 
 
 - major_index()[source]¶
- 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 - >>> from sage.all import * >>> Composition([Integer(1), Integer(1), Integer(3), Integer(1), Integer(2), Integer(1), Integer(3)]).major_index() 31 
 - meet(other, check=True)[source]¶
- Return the meet of - selfwith a composition- otherof 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 as- self
- check– boolean (default:- True); whether to check the input compositions for having the same size
 - OUTPUT: the meet of the compositions - selfand- other- 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] - >>> from sage.all import * >>> Composition([Integer(3), Integer(1), Integer(1), Integer(3), Integer(1)]).meet([Integer(4), Integer(3), Integer(2)]) [4, 5] >>> Composition([Integer(9), Integer(6)]).meet([Integer(1), Integer(3), Integer(6), Integer(3), Integer(2)]) [15] >>> Composition([Integer(9), Integer(6)]).meet([Integer(1), Integer(3), Integer(5), Integer(1), Integer(3), Integer(2)]) [9, 6] >>> Composition([Integer(1), Integer(1), Integer(1), Integer(1), Integer(1)]).meet([Integer(3), Integer(2)]) [3, 2] >>> Composition([Integer(4), Integer(2)]).meet([Integer(3), Integer(3)]) [6] >>> Composition([]).meet([]) [] >>> Composition([Integer(1)]).meet([Integer(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 - >>> from sage.all import * >>> all( all( I.is_finer(I.meet(J)) and ... J.is_finer(I.meet(J)) ... for J in Compositions(Integer(4)) ) ... for I in Compositions(Integer(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 - >>> from sage.all import * >>> all( all( all( I.meet(J).is_finer(K) ... for K in I.fatter() ... if J.is_finer(K) ) ... for J in Compositions(Integer(3)) ) ... for I in Compositions(Integer(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 - >>> from sage.all import * >>> 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) ) >>> all( test_meet(n) for n in range(Integer(1), Integer(5)) ) True - See also - AUTHORS: - Darij Grinberg (2013-09-05) 
 
 - near_concatenation(other)[source]¶
- Return the near-concatenation of two nonempty compositions - selfand- other.- 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 - Noneif 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] - >>> from sage.all import * >>> Composition([Integer(1), Integer(1), Integer(3)]).near_concatenation(Composition([Integer(4), Integer(1), Integer(2)])) [1, 1, 7, 1, 2] >>> Composition([Integer(6)]).near_concatenation(Composition([Integer(1), Integer(5)])) [7, 5] >>> Composition([Integer(1), Integer(5)]).near_concatenation(Composition([Integer(6)])) [1, 11] 
 - partial_sums(final=True)[source]¶
- 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– boolean (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] - >>> from sage.all import * >>> Composition([Integer(1),Integer(1),Integer(3),Integer(1),Integer(2),Integer(1),Integer(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] - >>> from sage.all import * >>> Composition([Integer(1),Integer(1),Integer(3),Integer(1),Integer(2),Integer(1),Integer(3)]).partial_sums(final=False) [1, 2, 5, 6, 8, 9] 
 - peaks()[source]¶
- 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] - >>> from sage.all import * >>> Composition([Integer(1), Integer(1), Integer(3), Integer(1), Integer(2), Integer(1), Integer(3)]).peaks() [4, 7] 
 - refinement_splitting(J)[source]¶
- Return the refinement splitting of - selfaccording to- J.- INPUT: - J– a composition such that- selfis finer than- J
 - 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]) - >>> from sage.all import * >>> Composition([Integer(1),Integer(2),Integer(2),Integer(1),Integer(1),Integer(2)]).refinement_splitting([Integer(5),Integer(1),Integer(3)]) [[1, 2, 2], [1], [1, 2]] >>> Composition([]).refinement_splitting([]) [] >>> Composition([Integer(3)]).refinement_splitting([Integer(2)]) Traceback (most recent call last): ... ValueError: compositions self (= [3]) and J (= [2]) must be of the same size >>> Composition([Integer(2),Integer(1)]).refinement_splitting([Integer(1),Integer(2)]) Traceback (most recent call last): ... ValueError: composition J (= [2, 1]) does not refine self (= [1, 2]) 
 - refinement_splitting_lengths(J)[source]¶
- Return the lengths of the compositions in the refinement splitting of - selfaccording to- J.- See also - refinement_splitting()for the definition of refinement splitting- EXAMPLES: - 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]) - >>> from sage.all import * >>> Composition([Integer(1),Integer(2),Integer(2),Integer(1),Integer(1),Integer(2)]).refinement_splitting_lengths([Integer(5),Integer(1),Integer(3)]) [3, 1, 2] >>> Composition([]).refinement_splitting_lengths([]) [] >>> Composition([Integer(3)]).refinement_splitting_lengths([Integer(2)]) Traceback (most recent call last): ... ValueError: compositions self (= [3]) and J (= [2]) must be of the same size >>> Composition([Integer(2),Integer(1)]).refinement_splitting_lengths([Integer(1),Integer(2)]) Traceback (most recent call last): ... ValueError: composition J (= [2, 1]) does not refine self (= [1, 2]) 
 - reversed()[source]¶
- 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] - >>> from sage.all import * >>> Composition([Integer(1), Integer(1), Integer(3), Integer(1), Integer(2), Integer(1), Integer(3)]).reversed() [3, 1, 2, 1, 3, 1, 1] 
 - ribbon_decomposition(other, check=True)[source]¶
- Return a pair describing the ribbon decomposition of a composition - selfwith respect to a composition- otherof 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__()and- near_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 as- self
- check– boolean (default:- True); whether to check the input compositions for having the same size
 - OUTPUT: - a pair - (u, v), where- uis a tuple of compositions (corresponding to the \(m\)-tuple \((I_1, I_2, \ldots , I_m)\) in the above definition), and- vis a tuple of \(0`s and `1`s (encoding the choice of signs `\bullet\) 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([]) ((), ()) - >>> from sage.all import * >>> Composition([Integer(3), Integer(1), Integer(1), Integer(3), Integer(1)]).ribbon_decomposition([Integer(4), Integer(3), Integer(2)]) (([3, 1], [1, 2], [1, 1]), (0, 1)) >>> Composition([Integer(9), Integer(6)]).ribbon_decomposition([Integer(1), Integer(3), Integer(6), Integer(3), Integer(2)]) (([1], [3], [5, 1], [3], [2]), (1, 1, 1, 1)) >>> Composition([Integer(9), Integer(6)]).ribbon_decomposition([Integer(1), Integer(3), Integer(5), Integer(1), Integer(3), Integer(2)]) (([1], [3], [5], [1], [3], [2]), (1, 1, 0, 1, 1)) >>> Composition([Integer(1), Integer(1), Integer(1), Integer(1), Integer(1)]).ribbon_decomposition([Integer(3), Integer(2)]) (([1, 1, 1], [1, 1]), (0,)) >>> Composition([Integer(4), Integer(2)]).ribbon_decomposition([Integer(6)]) (([4, 2],), ()) >>> 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 - >>> from sage.all import * >>> def compose_back(u, v): ... comp = u[Integer(0)] ... r = len(v) ... if len(u) != r + Integer(1): ... raise ValueError("something is wrong") ... for i in range(r): ... if v[i] == Integer(0): ... comp += u[i + Integer(1)] ... else: ... comp = comp.near_concatenation(u[i + Integer(1)]) ... return comp >>> all( all( all( compose_back(*(I.ribbon_decomposition(J))) == I ... for J in Compositions(n) ) ... for I in Compositions(n) ) ... for n in range(Integer(1), Integer(5)) ) True - AUTHORS: - Darij Grinberg (2013-08-29) 
 
 - shuffle_product(other, overlap=False)[source]¶
- The (overlapping) shuffles of - selfand- other.- 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– composition
- overlap– boolean (default:- False); if- True, 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 # needs sage.combinat Shuffle product of [2, 2] and [1, 1, 3] sage: S.list() # needs sage.combinat [[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]] - >>> from sage.all import * >>> alph = Composition([Integer(2),Integer(2)]) >>> beta = Composition([Integer(1),Integer(1),Integer(3)]) >>> S = alph.shuffle_product(beta); S # needs sage.combinat Shuffle product of [2, 2] and [1, 1, 3] >>> S.list() # needs sage.combinat [[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 # needs sage.combinat Overlapping shuffle product of [2, 2] and [1, 1, 3] sage: O.list() # needs sage.combinat [[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]] - >>> from sage.all import * >>> alph = Composition([Integer(2),Integer(2)]) >>> beta = Composition([Integer(1),Integer(1),Integer(3)]) >>> O = alph.shuffle_product(beta, overlap=True); O # needs sage.combinat Overlapping shuffle product of [2, 2] and [1, 1, 3] >>> O.list() # needs sage.combinat [[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: # needs sage.combinat 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]] - >>> from sage.all import * >>> # needs sage.combinat >>> w1 = Composition([Integer(1)]) >>> S = w1.shuffle_product(w1); S Shuffle product of [1] and [1] >>> S.list() [[1, 1], [1, 1]] >>> O = w1.shuffle_product(w1, overlap=True); O Overlapping shuffle product of [1] and [1] >>> O.list() [[1, 1], [1, 1], [2]] 
 - size()[source]¶
- Return the size of - self, that is the sum of its parts.- EXAMPLES: - sage: Composition([7,1,3]).size() 11 - >>> from sage.all import * >>> Composition([Integer(7),Integer(1),Integer(3)]).size() 11 
 - specht_module(base_ring=None)[source]¶
- Return the Specht module corresponding to - self.- EXAMPLES: - sage: SM = Composition([1,2,2]).specht_module(QQ); SM # needs sage.combinat sage.modules Specht module of [(0, 0), (1, 0), (1, 1), (2, 0), (2, 1)] over Rational Field sage: s = SymmetricFunctions(QQ).s() # needs sage.combinat sage.modules sage: s(SM.frobenius_image()) # needs sage.combinat sage.modules s[2, 2, 1] - >>> from sage.all import * >>> SM = Composition([Integer(1),Integer(2),Integer(2)]).specht_module(QQ); SM # needs sage.combinat sage.modules Specht module of [(0, 0), (1, 0), (1, 1), (2, 0), (2, 1)] over Rational Field >>> s = SymmetricFunctions(QQ).s() # needs sage.combinat sage.modules >>> s(SM.frobenius_image()) # needs sage.combinat sage.modules s[2, 2, 1] 
 - specht_module_dimension(base_ring=None)[source]¶
- Return the dimension of the Specht module corresponding to - self.- INPUT: - base_ring– (default: \(\QQ\)) the base ring
 - EXAMPLES: - sage: Composition([1,2,2]).specht_module_dimension() # needs sage.combinat sage.modules 5 sage: Composition([1,2,2]).specht_module_dimension(GF(2)) # needs sage.combinat sage.modules sage.rings.finite_rings 5 - >>> from sage.all import * >>> Composition([Integer(1),Integer(2),Integer(2)]).specht_module_dimension() # needs sage.combinat sage.modules 5 >>> Composition([Integer(1),Integer(2),Integer(2)]).specht_module_dimension(GF(Integer(2))) # needs sage.combinat sage.modules sage.rings.finite_rings 5 
 - static sum(compositions)[source]¶
- Return the concatenation of the given compositions. - INPUT: - compositions– 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] - >>> from sage.all import * >>> Composition.sum([Composition([Integer(1), Integer(1), Integer(3)]), Composition([Integer(4), Integer(1), Integer(2)]), Composition([Integer(3),Integer(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] - >>> from sage.all import * >>> Composition.sum([Composition([i,i]) for i in [Integer(4),Integer(1),Integer(3)]]) [4, 4, 1, 1, 3, 3] - Empty inputs are handled gracefully: - sage: Composition.sum([]) == Composition([]) True - >>> from sage.all import * >>> Composition.sum([]) == Composition([]) True 
 - sup(other, check=True)[source]¶
- Return the join of - selfwith a composition- otherof 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 as- self
- check– boolean (default:- True); whether to check the input compositions for having the same size
 - OUTPUT: the join of the compositions - selfand- other- 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([]) [] - >>> from sage.all import * >>> Composition([Integer(3), Integer(1), Integer(1), Integer(3), Integer(1)]).join([Integer(4), Integer(3), Integer(2)]) [3, 1, 1, 2, 1, 1] >>> Composition([Integer(9), Integer(6)]).join([Integer(1), Integer(3), Integer(6), Integer(3), Integer(2)]) [1, 3, 5, 1, 3, 2] >>> Composition([Integer(9), Integer(6)]).join([Integer(1), Integer(3), Integer(5), Integer(1), Integer(3), Integer(2)]) [1, 3, 5, 1, 3, 2] >>> Composition([Integer(1), Integer(1), Integer(1), Integer(1), Integer(1)]).join([Integer(3), Integer(2)]) [1, 1, 1, 1, 1] >>> Composition([Integer(4), Integer(2)]).join([Integer(3), Integer(3)]) [3, 1, 2] >>> 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 - >>> from sage.all import * >>> all( all( I.join(J).is_finer(I) and ... I.join(J).is_finer(J) ... for J in Compositions(Integer(4)) ) ... for I in Compositions(Integer(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 - >>> from sage.all import * >>> all( all( all( K.is_finer(I.join(J)) ... for K in I.finer() ... if K.is_finer(J) ) ... for J in Compositions(Integer(3)) ) ... for I in Compositions(Integer(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 - >>> from sage.all import * >>> all( all( Composition.sum(I.ribbon_decomposition(J)[Integer(0)]) ... == I.join(J) for J in Compositions(Integer(4)) ) ... for I in Compositions(Integer(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 - >>> from sage.all import * >>> all( all( I.to_subset().union(J.to_subset()) ... == I.join(J).to_subset() ... for J in Compositions(Integer(4)) ) ... for I in Compositions(Integer(4)) ) True - See also - AUTHORS: - Darij Grinberg (2013-09-05) 
 
 - to_code()[source]¶
- 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] - >>> from sage.all import * >>> Composition([Integer(4),Integer(1),Integer(2),Integer(3),Integer(5)]).to_code() [1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0] 
 - to_partition()[source]¶
- Return the partition obtained by sorting - selfinto decreasing order.- EXAMPLES: - sage: Composition([2,1,3]).to_partition() # needs sage.combinat [3, 2, 1] sage: Composition([4,2,2]).to_partition() # needs sage.combinat [4, 2, 2] sage: Composition([]).to_partition() # needs sage.combinat [] - >>> from sage.all import * >>> Composition([Integer(2),Integer(1),Integer(3)]).to_partition() # needs sage.combinat [3, 2, 1] >>> Composition([Integer(4),Integer(2),Integer(2)]).to_partition() # needs sage.combinat [4, 2, 2] >>> Composition([]).to_partition() # needs sage.combinat [] 
 - to_skew_partition(overlap=1)[source]¶
- Return the skew partition obtained from - self.- This is a skew partition whose rows have the entries of - selfas their length, taken in reverse order (so the first entry of- selfis the length of the lowermost row, etc.). The parameter- overlapindicates 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 of- self.- EXAMPLES: - sage: # needs sage.combinat 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] - >>> from sage.all import * >>> # needs sage.combinat >>> Composition([Integer(3),Integer(4),Integer(1)]).to_skew_partition() [6, 6, 3] / [5, 2] >>> Composition([Integer(3),Integer(4),Integer(1)]).to_skew_partition(overlap=Integer(0)) [8, 7, 3] / [7, 3] >>> Composition([]).to_skew_partition() [] / [] >>> Composition([Integer(1),Integer(2)]).to_skew_partition() [2, 1] / [] >>> Composition([Integer(2),Integer(1)]).to_skew_partition() [2, 2] / [1] 
 - to_subset(final=False)[source]¶
- The subset corresponding to - selfunder 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– boolean (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} {} - >>> from sage.all import * >>> Composition([Integer(1),Integer(1),Integer(3),Integer(1),Integer(2),Integer(1),Integer(3)]).to_subset() {1, 2, 5, 6, 8, 9} >>> for I in Compositions(Integer(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} - >>> from sage.all import * >>> Composition([Integer(1),Integer(1),Integer(3),Integer(1),Integer(2),Integer(1),Integer(3)]).to_subset(final=True) {1, 2, 5, 6, 8, 9, 12} 
 - wll_gt(co2)[source]¶
- Return - Trueif the composition- selfis greater than the composition- co2with respect to the wll-ordering; otherwise, return- False.- 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 - >>> from sage.all import * >>> Composition([Integer(4),Integer(1),Integer(2)]).wll_gt([Integer(3),Integer(1),Integer(3)]) True >>> Composition([Integer(7)]).wll_gt([Integer(4),Integer(1),Integer(2)]) False >>> Composition([Integer(8)]).wll_gt([Integer(4),Integer(1),Integer(2)]) True >>> Composition([Integer(3),Integer(2),Integer(2),Integer(2)]).wll_gt([Integer(5),Integer(2)]) True >>> Composition([]).wll_gt([Integer(3)]) False >>> Composition([Integer(2),Integer(1)]).wll_gt([Integer(2),Integer(1)]) False >>> Composition([Integer(2),Integer(2),Integer(2)]).wll_gt([Integer(4),Integer(2)]) True >>> Composition([Integer(4),Integer(2)]).wll_gt([Integer(2),Integer(2),Integer(2)]) False >>> Composition([Integer(1),Integer(1),Integer(2)]).wll_gt([Integer(2),Integer(2)]) True >>> Composition([Integer(2),Integer(2)]).wll_gt([Integer(1),Integer(3)]) True >>> Composition([Integer(2),Integer(1),Integer(2)]).wll_gt([]) True 
 
- class sage.combinat.composition.Compositions(is_infinite=False, category=None)[source]¶
- Bases: - UniqueRepresentation,- 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 - >>> from sage.all import * >>> Compositions(Integer(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]] - >>> from sage.all import * >>> Compositions(Integer(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] - >>> from sage.all import * >>> Compositions(Integer(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] - >>> from sage.all import * >>> Compositions(Integer(4)).next([Integer(1),Integer(1),Integer(2)]) [1, 2, 1] - If \(n\) is not specified, this returns the combinatorial class of all (nonnegative) integer compositions: - sage: Compositions() Compositions of nonnegative integers sage: [] in Compositions() True sage: [2,3,1] in Compositions() True sage: [-2,3,1] in Compositions() False - >>> from sage.all import * >>> Compositions() Compositions of nonnegative integers >>> [] in Compositions() True >>> [Integer(2),Integer(3),Integer(1)] in Compositions() True >>> [-Integer(2),Integer(3),Integer(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 - >>> from sage.all import * >>> Compositions(Integer(3)) Compositions of 3 >>> list(Compositions(Integer(3))) [[1, 1, 1], [1, 2], [2, 1], [3]] >>> Compositions(Integer(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 - >>> from sage.all import * >>> [Integer(3),Integer(4)] in Compositions() True >>> [Integer(3),Integer(4)] in Compositions(Integer(7)) True >>> [Integer(3),Integer(4)] in Compositions(Integer(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 - >>> from sage.all import * >>> [Integer(4),Integer(2)] in Compositions(Integer(6), inner=[Integer(2),Integer(2)]) True >>> [Integer(4),Integer(2)] in Compositions(Integer(6), inner=[Integer(2),Integer(3)]) False >>> [Integer(4),Integer(1)] in Compositions(Integer(5), inner=[Integer(2),Integer(1)], max_slope = Integer(0)) True - An example with incompatible constraints: - sage: [4,2] in Compositions(6, inner=[2,2], min_part=3) False - >>> from sage.all import * >>> [Integer(4),Integer(2)] in Compositions(Integer(6), inner=[Integer(2),Integer(2)], min_part=Integer(3)) False - The options - length,- min_length, and- max_lengthcan 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]] - >>> from sage.all import * >>> Compositions(Integer(4), length=Integer(2)).list() [[3, 1], [2, 2], [1, 3]] >>> Compositions(Integer(4), min_length=Integer(2)).list() [[3, 1], [2, 2], [2, 1, 1], [1, 3], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]] >>> Compositions(Integer(4), max_length=Integer(2)).list() [[4], [3, 1], [2, 2], [1, 3]] - Setting both - min_lengthand- max_lengthto the same value is equivalent to setting- lengthto this value:- sage: Compositions(4, min_length=2, max_length=2).list() [[3, 1], [2, 2], [1, 3]] - >>> from sage.all import * >>> Compositions(Integer(4), min_length=Integer(2), max_length=Integer(2)).list() [[3, 1], [2, 2], [1, 3]] - The options - innerand- outercan 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]] - >>> from sage.all import * >>> list(Compositions(Integer(4), outer=[Integer(3),Integer(1),Integer(2)])) [[3, 1], [2, 1, 1], [1, 1, 2]] - outersets- max_lengthto the length of its argument. Moreover, the parts of- outermay 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]] - >>> from sage.all import * >>> Compositions(Integer(4), outer=[Integer(1),oo,Integer(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]] - >>> from sage.all import * >>> Compositions(Integer(4), inner=[Integer(1),Integer(1),Integer(1)]).list() [[2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]] - The options - min_slopeand- max_slopecan be used to set constraints on the slope, that is the difference- p[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]] - >>> from sage.all import * >>> Compositions(Integer(4), min_slope=Integer(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]] - >>> from sage.all import * >>> Compositions(Integer(4), max_slope=Integer(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]] - >>> from sage.all import * >>> Compositions(Integer(4), min_slope=-Integer(1), max_slope=Integer(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]] - >>> from sage.all import * >>> Compositions(Integer(5), max_slope=Integer(1), min_slope=-Integer(2), min_length=Integer(2), max_length=Integer(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]] - >>> from sage.all import * >>> Compositions(Integer(5), max_slope=Integer(1), min_slope=-Integer(2), min_length=Integer(2), max_length=Integer(4), outer=[Integer(2),Integer(5),Integer(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 setting - min_part=0is not allowed:- sage: Compositions(2, length=3, min_part=0) Traceback (most recent call last): ... ValueError: setting min_part=0 is not allowed for Compositions - >>> from sage.all import * >>> Compositions(Integer(2), length=Integer(3), min_part=Integer(0)) Traceback (most recent call last): ... ValueError: setting min_part=0 is not allowed for Compositions - Instead you must use - IntegerVectors:- sage: list(IntegerVectors(2, 3)) [[2, 0, 0], [1, 1, 0], [1, 0, 1], [0, 2, 0], [0, 1, 1], [0, 0, 2]] - >>> from sage.all import * >>> list(IntegerVectors(Integer(2), Integer(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[source]¶
- alias of - Composition
 - from_code(code)[source]¶
- 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 sage.all import * >>> Composition([Integer(4),Integer(1),Integer(2),Integer(3),Integer(5)]).to_code() [1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0] >>> Compositions().from_code(_) [4, 1, 2, 3, 5] >>> Composition([Integer(3),Integer(1),Integer(2),Integer(3),Integer(5)]).to_code() [1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0] >>> Compositions().from_code(_) [3, 1, 2, 3, 5] 
 - from_descents(descents, nps=None)[source]¶
- Return a composition from the list of descents. - INPUT: - descents– an iterable
- nps– integer or- None(default:- None)
 - OUTPUT: - The composition of - npswhose descents are listed in- descents, assuming that- npsis not- None(otherwise, the last element of- descentsis removed from- descents, and- npsis 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 sage.all import * >>> [x-Integer(1) for x in Composition([Integer(1), Integer(1), Integer(3), Integer(4), Integer(3)]).to_subset()] [0, 1, 4, 8] >>> Compositions().from_descents([Integer(1),Integer(0),Integer(4),Integer(8)],Integer(12)) [1, 1, 3, 4, 3] >>> Compositions().from_descents([Integer(1),Integer(0),Integer(4),Integer(8),Integer(11)]) [1, 1, 3, 4, 3] 
 - from_subset(S, n)[source]¶
- The composition of \(n\) corresponding to the subset - Sof \(\{1, 2, \ldots, n-1\}\) 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- Composition.to_subset()).- INPUT: - S– an iterable, a subset of \(\{1, 2, \ldots, n-1\}\)
- n– 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) [] - >>> from sage.all import * >>> Compositions().from_subset([Integer(2),Integer(1),Integer(5),Integer(9)], Integer(12)) [1, 1, 3, 4, 3] >>> Compositions().from_subset({Integer(2),Integer(1),Integer(5),Integer(9)}, Integer(12)) [1, 1, 3, 4, 3] >>> Compositions().from_subset([], Integer(12)) [12] >>> Compositions().from_subset([], Integer(0)) [] 
 
- class sage.combinat.composition.Compositions_all[source]¶
- Bases: - Compositions- Class of all compositions. - subset(size=None)[source]¶
- 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 - >>> from sage.all import * >>> C = Compositions() >>> C.subset(Integer(4)) Compositions of 4 >>> C.subset(size=Integer(3)) Compositions of 3 
 
- class sage.combinat.composition.Compositions_constraints(*args, **kwds)[source]¶
- Bases: - IntegerListsLex
- class sage.combinat.composition.Compositions_n(n)[source]¶
- Bases: - Compositions- Class of compositions of a fixed \(n\). - random_element()[source]¶
- Return a random - Compositionwith 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] - >>> from sage.all import * >>> Compositions(Integer(5)).random_element() # random [2, 1, 1, 1] >>> Compositions(Integer(0)).random_element() [] >>> Compositions(Integer(1)).random_element() [1]