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:
Bases: sage.combinat.combinat.CombinatorialObject, sage.structure.element.Element
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]
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]
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]
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
TESTS:
sage: parent(list(Compositions(1))[0].conjugate())
Compositions of 1
sage: parent(list(Compositions(0))[0].conjugate())
Compositions of 0
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:
OUTPUT:
EXAMPLES:
sage: c = Composition([2,1,3,2])
sage: c.descents()
[1, 2, 5]
sage: c.descents(final_descent=True)
[1, 2, 5, 7]
Return the composition fatter than self, obtained by grouping together consecutive parts according to grouping.
INPUT:
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]
TESTS:
sage: Composition([]).fatten(Composition([]))
[]
sage: c.fatten(Composition([3,1,1])).__class__ == c.__class__
True
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
Return the set of compositions which are finer than self.
EXAMPLES:
sage: C = Composition([3,2]).finer()
sage: C.cardinality()
8
sage: list(C)
[[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]]
Return the meet of self with a composition other 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:
OUTPUT:
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
sage: all( test_meet(n) for n in range(5, 9) ) # long time
True
TESTS:
sage: Composition([3, 1, 1, 3, 1]).meet([4, 3, 1])
Traceback (most recent call last):
...
ValueError: [3, 1, 1, 3, 1] is not the same size as [4, 3, 1]
See also
AUTHORS:
Return True if the composition self is 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
Return the join of self with a composition other 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:
OUTPUT:
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 conctenation 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
TESTS:
sage: Composition([3, 1, 1, 3, 1]).join([4, 3, 1])
Traceback (most recent call last):
...
ValueError: [3, 1, 1, 3, 1] is not the same size as [4, 3, 1]
See also
AUTHORS:
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
Return the meet of self with a composition other 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:
OUTPUT:
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
sage: all( test_meet(n) for n in range(5, 9) ) # long time
True
TESTS:
sage: Composition([3, 1, 1, 3, 1]).meet([4, 3, 1])
Traceback (most recent call last):
...
ValueError: [3, 1, 1, 3, 1] is not the same size as [4, 3, 1]
See also
AUTHORS:
Return the near-concatenation of two nonempty compositions self and 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 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]
TESTS:
sage: Composition([]).near_concatenation(Composition([]))
sage: Composition([]).near_concatenation(Composition([2, 1]))
sage: Composition([3, 2]).near_concatenation(Composition([]))
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:
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]
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]
Return the refinement splitting of self according to J.
INPUT:
OUTPUT:
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])
Return the lengths of the compositions in the refinement splitting of self according 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])
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]
Return a pair describing the ribbon decomposition of a composition self with respect to a composition other 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__() 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:
OUTPUT:
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
TESTS:
sage: Composition([3, 1, 1, 3, 1]).ribbon_decomposition([4, 3, 1])
Traceback (most recent call last):
...
ValueError: [3, 1, 1, 3, 1] is not the same size as [4, 3, 1]
AUTHORS:
The (overlapping) shuffles of self and 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:
OUTPUT:
An enumerated set (allowing for mutliplicities)
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: S = Composition([1]).shuffle_product([1]); S
Shuffle product of [1] and [1]
sage: S.list()
[[1, 1], [1, 1]]
sage: O = Composition([1]).shuffle_product([1], overlap=True); O
Overlapping shuffle product of [1] and [1]
sage: O.list()
[[1, 1], [1, 1], [2]]
TESTS:
sage: Composition([]).shuffle_product([]).list()
[[]]
Return the size of self, that is the sum of its parts.
EXAMPLES:
sage: Composition([7,1,3]).size()
11
Return the concatenation of the given compositions.
INPUT:
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
Return the join of self with a composition other 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:
OUTPUT:
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 conctenation 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
TESTS:
sage: Composition([3, 1, 1, 3, 1]).join([4, 3, 1])
Traceback (most recent call last):
...
ValueError: [3, 1, 1, 3, 1] is not the same size as [4, 3, 1]
See also
AUTHORS:
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]
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()
[]
Return the skew partition obtained from self. This is a skew partition whose rows have the entries of self as their length, taken in reverse order (so the first entry of self is the length of the lowermost row, etc.). The parameter overlap 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 of self.
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]
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:
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}
TESTS:
We verify that to_subset is indeed a bijection for compositions of size \(n = 8\):
sage: n = 8
sage: all(Composition(from_subset=(S, n)).to_subset() == S \
... for S in Subsets(n-1))
True
sage: all(Composition(from_subset=(I.to_subset(), n)) == I \
... for I in Compositions(n))
True
Return True if the composition self is greater than the composition co2 with 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:
(“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
Bases: sage.structure.parent.Parent, sage.structure.unique_representation.UniqueRepresentation
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
Note that the given constraints should be compatible:
sage: [4,2] in Compositions(6, inner=[2,2], min_part=3)
True
The options length, min_length, and max_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 and max_length to the same value is equivalent to setting length to this value:
sage: Compositions(4, min_length=2, max_length=2).list()
[[3, 1], [2, 2], [1, 3]]
The options inner and outer 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 sets max_length to the length of its argument. Moreover, the parts of outer 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 and max_slope can 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]]
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 using IntegerVectors 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.
TESTS:
sage: C = Compositions(4, length=2)
sage: C == loads(dumps(C))
True
sage: Compositions(6, min_part=2, length=3)
Compositions of the integer 6 satisfying constraints length=3, min_part=2
sage: [2, 1] in Compositions(3, length=2)
True
sage: [2,1,2] in Compositions(5, min_part=1)
True
sage: [2,1,2] in Compositions(5, min_part=2)
False
sage: Compositions(4, length=2).cardinality()
3
sage: Compositions(4, min_length=2).cardinality()
7
sage: Compositions(4, max_length=2).cardinality()
4
sage: Compositions(4, max_part=2).cardinality()
5
sage: Compositions(4, min_part=2).cardinality()
2
sage: Compositions(4, outer=[3,1,2]).cardinality()
3
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]]
sage: Compositions(4, max_part=2).list()
[[2, 2], [2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]
sage: Compositions(4, min_part=2).list()
[[4], [2, 2]]
sage: Compositions(4, outer=[3,1,2]).list()
[[3, 1], [2, 1, 1], [1, 1, 2]]
sage: Compositions(3, outer = Composition([3,2])).list()
[[3], [2, 1], [1, 2]]
sage: Compositions(4, outer=[1,oo,1]).list()
[[1, 3], [1, 2, 1]]
sage: Compositions(4, inner=[1,1,1]).list()
[[2, 1, 1], [1, 2, 1], [1, 1, 2], [1, 1, 1, 1]]
sage: Compositions(4, inner=Composition([1,2])).list()
[[2, 2], [1, 3], [1, 2, 1]]
sage: Compositions(4, min_slope=0).list()
[[4], [2, 2], [1, 3], [1, 1, 2], [1, 1, 1, 1]]
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]]
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]]
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]]
alias of Composition
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]
Return a composition from the list of descents.
INPUT:
OUTPUT:
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]
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 \(\{i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + \cdots + i_{k-1}\}\) (see Composition.to_subset()).
INPUT:
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)
[]
TESTS:
sage: Compositions().from_subset([2,1,5,9],9)
Traceback (most recent call last):
...
ValueError: S (=[1, 2, 5, 9]) is not a subset of {1, ..., 8}
Bases: sage.combinat.composition.Compositions
Class of all compositions.
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
Bases: sage.combinat.integer_list.IntegerListsLex
Initialize self.
TESTS:
sage: C = IntegerListsLex(2, length=3)
sage: C == loads(dumps(C))
True
sage: C == loads(dumps(C)) # this did fail at some point, really!
True
sage: C is loads(dumps(C)) # todo: not implemented
True
sage: C.cardinality().parent() is ZZ
True
sage: TestSuite(C).run()
Bases: sage.combinat.composition.Compositions
Class of compositions of a fixed \(n\).
Return the number of compositions of \(n\).
TESTS:
sage: Compositions(3).cardinality()
4
sage: Compositions(0).cardinality()
1
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]
TESTS:
sage: all([Compositions(10).random_element() in Compositions(10) for i in range(20)])
True