Partition/Diagram Algebras

class sage.combinat.partition_algebra.PartitionAlgebraElement_ak

Bases: PartitionAlgebraElement_generic

class sage.combinat.partition_algebra.PartitionAlgebraElement_bk

Bases: PartitionAlgebraElement_generic

class sage.combinat.partition_algebra.PartitionAlgebraElement_generic

Bases: IndexedFreeModuleElement

class sage.combinat.partition_algebra.PartitionAlgebraElement_pk

Bases: PartitionAlgebraElement_generic

class sage.combinat.partition_algebra.PartitionAlgebraElement_prk

Bases: PartitionAlgebraElement_generic

class sage.combinat.partition_algebra.PartitionAlgebraElement_rk

Bases: PartitionAlgebraElement_generic

class sage.combinat.partition_algebra.PartitionAlgebraElement_sk

Bases: PartitionAlgebraElement_generic

class sage.combinat.partition_algebra.PartitionAlgebraElement_tk

Bases: PartitionAlgebraElement_generic

class sage.combinat.partition_algebra.PartitionAlgebra_ak(R, k, n, name=None)

Bases: PartitionAlgebra_generic

EXAMPLES:

sage: from sage.combinat.partition_algebra import *
sage: p = PartitionAlgebra_ak(QQ, 3, 1)
sage: p == loads(dumps(p))
True

class sage.combinat.partition_algebra.PartitionAlgebra_bk(R, k, n, name=None)

Bases: PartitionAlgebra_generic

EXAMPLES:

sage: from sage.combinat.partition_algebra import *
sage: p = PartitionAlgebra_bk(QQ, 3, 1)
sage: p == loads(dumps(p))
True

class sage.combinat.partition_algebra.PartitionAlgebra_generic(R, cclass, n, k, name=None, prefix=None)

Bases: CombinatorialFreeModule

EXAMPLES:

sage: from sage.combinat.partition_algebra import *
sage: s = PartitionAlgebra_sk(QQ, 3, 1)
sage: TestSuite(s).run()
sage: s == loads(dumps(s))
True

one_basis()

Return the basis index for the unit of the algebra.

EXAMPLES:

sage: from sage.combinat.partition_algebra import *
sage: s = PartitionAlgebra_sk(ZZ, 3, 1)
sage: len(s.one().support())   # indirect doctest
1

product_on_basis(left, right)

EXAMPLES:

sage: from sage.combinat.partition_algebra import *
sage: s = PartitionAlgebra_sk(QQ, 3, 1)
sage: t12 = s(Set([Set([1,-2]),Set([2,-1]),Set([3,-3])]))
sage: t12^2 == s(1) #indirect doctest
True

class sage.combinat.partition_algebra.PartitionAlgebra_pk(R, k, n, name=None)

Bases: PartitionAlgebra_generic

EXAMPLES:

sage: from sage.combinat.partition_algebra import *
sage: p = PartitionAlgebra_pk(QQ, 3, 1)
sage: p == loads(dumps(p))
True

class sage.combinat.partition_algebra.PartitionAlgebra_prk(R, k, n, name=None)

Bases: PartitionAlgebra_generic

EXAMPLES:

sage: from sage.combinat.partition_algebra import *
sage: p = PartitionAlgebra_prk(QQ, 3, 1)
sage: p == loads(dumps(p))
True

class sage.combinat.partition_algebra.PartitionAlgebra_rk(R, k, n, name=None)

Bases: PartitionAlgebra_generic

EXAMPLES:

sage: from sage.combinat.partition_algebra import *
sage: p = PartitionAlgebra_rk(QQ, 3, 1)
sage: p == loads(dumps(p))
True

class sage.combinat.partition_algebra.PartitionAlgebra_sk(R, k, n, name=None)

Bases: PartitionAlgebra_generic

EXAMPLES:

sage: from sage.combinat.partition_algebra import *
sage: p = PartitionAlgebra_sk(QQ, 3, 1)
sage: p == loads(dumps(p))
True

class sage.combinat.partition_algebra.PartitionAlgebra_tk(R, k, n, name=None)

Bases: PartitionAlgebra_generic

EXAMPLES:

sage: from sage.combinat.partition_algebra import *
sage: p = PartitionAlgebra_tk(QQ, 3, 1)
sage: p == loads(dumps(p))
True

sage.combinat.partition_algebra.SetPartitionsAk(k)

Return the combinatorial class of set partitions of type \(A_k\).

EXAMPLES:

sage: A3 = SetPartitionsAk(3); A3
Set partitions of {1, ..., 3, -1, ..., -3}

sage: A3.first() #random
{{1, 2, 3, -1, -3, -2}}
sage: A3.last() #random
{{-1}, {-2}, {3}, {1}, {-3}, {2}}
sage: A3.random_element()  #random
{{1, 3, -3, -1}, {2, -2}}

sage: A3.cardinality()
203

sage: A2p5 = SetPartitionsAk(2.5); A2p5
Set partitions of {1, ..., 3, -1, ..., -3} with 3 and -3 in the same block
sage: A2p5.cardinality()
52

sage: A2p5.first() #random
{{1, 2, 3, -1, -3, -2}}
sage: A2p5.last() #random
{{-1}, {-2}, {2}, {3, -3}, {1}}
sage: A2p5.random_element() #random
{{-1}, {-2}, {3, -3}, {1, 2}}

class sage.combinat.partition_algebra.SetPartitionsAk_k(k)

Bases: SetPartitions_set

Element

alias of SetPartitionsXkElement

class sage.combinat.partition_algebra.SetPartitionsAkhalf_k(k)

Bases: SetPartitions_set

Element

alias of SetPartitionsXkElement

sage.combinat.partition_algebra.SetPartitionsBk(k)

Return the combinatorial class of set partitions of type \(B_k\).

These are the set partitions where every block has size 2.

EXAMPLES:

sage: B3 = SetPartitionsBk(3); B3
Set partitions of {1, ..., 3, -1, ..., -3} with block size 2

sage: B3.first() #random
{{2, -2}, {1, -3}, {3, -1}}
sage: B3.last() #random
{{1, 2}, {3, -2}, {-3, -1}}
sage: B3.random_element() #random
{{2, -1}, {1, -3}, {3, -2}}

sage: B3.cardinality()
15

sage: B2p5 = SetPartitionsBk(2.5); B2p5
Set partitions of {1, ..., 3, -1, ..., -3} with 3 and -3 in the same block and with block size 2

sage: B2p5.first() #random
{{2, -1}, {3, -3}, {1, -2}}
sage: B2p5.last() #random
{{1, 2}, {3, -3}, {-1, -2}}
sage: B2p5.random_element() #random
{{2, -2}, {3, -3}, {1, -1}}

sage: B2p5.cardinality()
3

class sage.combinat.partition_algebra.SetPartitionsBk_k(k)

Bases: SetPartitionsAk_k

cardinality()

Return the number of set partitions in \(B_k\) where \(k\) is an integer.

This is given by (2k)!! = (2k-1)*(2k-3)*…*5*3*1.

EXAMPLES:

sage: SetPartitionsBk(3).cardinality()
15
sage: SetPartitionsBk(2).cardinality()
3
sage: SetPartitionsBk(1).cardinality()
1
sage: SetPartitionsBk(4).cardinality()
105
sage: SetPartitionsBk(5).cardinality()
945

class sage.combinat.partition_algebra.SetPartitionsBkhalf_k(k)

Bases: SetPartitionsAkhalf_k

cardinality()

sage.combinat.partition_algebra.SetPartitionsIk(k)

Return the combinatorial class of set partitions of type \(I_k\).

These are set partitions with a propagating number of less than \(k\). Note that the identity set partition \(\{\{1, -1\}, \ldots, \{k, -k\}\}\) is not in \(I_k\).

EXAMPLES:

sage: I3 = SetPartitionsIk(3); I3
Set partitions of {1, ..., 3, -1, ..., -3} with propagating number < 3
sage: I3.cardinality()
197

sage: I3.first() #random
{{1, 2, 3, -1, -3, -2}}
sage: I3.last() #random
{{-1}, {-2}, {3}, {1}, {-3}, {2}}
sage: I3.random_element() #random
{{-1}, {-3, -2}, {2, 3}, {1}}

sage: I2p5 = SetPartitionsIk(2.5); I2p5
Set partitions of {1, ..., 3, -1, ..., -3} with 3 and -3 in the same block and propagating number < 3
sage: I2p5.cardinality()
50

sage: I2p5.first() #random
{{1, 2, 3, -1, -3, -2}}
sage: I2p5.last() #random
{{-1}, {-2}, {2}, {3, -3}, {1}}
sage: I2p5.random_element() #random
{{-1}, {-2}, {1, 3, -3}, {2}}

class sage.combinat.partition_algebra.SetPartitionsIk_k(k)

Bases: SetPartitionsAk_k

cardinality()

class sage.combinat.partition_algebra.SetPartitionsIkhalf_k(k)

Bases: SetPartitionsAkhalf_k

cardinality()

sage.combinat.partition_algebra.SetPartitionsPRk(k)

Return the combinatorial class of set partitions of type \(PR_k\).

EXAMPLES:

sage: SetPartitionsPRk(3)
Set partitions of {1, ..., 3, -1, ..., -3} with at most 1 positive
 and negative entry in each block and that are planar

class sage.combinat.partition_algebra.SetPartitionsPRk_k(k)

Bases: SetPartitionsRk_k

cardinality()

class sage.combinat.partition_algebra.SetPartitionsPRkhalf_k(k)

Bases: SetPartitionsRkhalf_k

cardinality()

sage.combinat.partition_algebra.SetPartitionsPk(k)

Return the combinatorial class of set partitions of type \(P_k\).

These are the planar set partitions.

EXAMPLES:

sage: P3 = SetPartitionsPk(3); P3
Set partitions of {1, ..., 3, -1, ..., -3} that are planar
sage: P3.cardinality()
132

sage: P3.first() #random
{{1, 2, 3, -1, -3, -2}}
sage: P3.last() #random
{{-1}, {-2}, {3}, {1}, {-3}, {2}}
sage: P3.random_element() #random
{{1, 2, -1}, {-3}, {3, -2}}

sage: P2p5 = SetPartitionsPk(2.5); P2p5
Set partitions of {1, ..., 3, -1, ..., -3} with 3 and -3 in the same block and that are planar
sage: P2p5.cardinality()
42

sage: P2p5.first() #random
{{1, 2, 3, -1, -3, -2}}
sage: P2p5.last() #random
{{-1}, {-2}, {2}, {3, -3}, {1}}
sage: P2p5.random_element() #random
{{1, 2, 3, -3}, {-1, -2}}

class sage.combinat.partition_algebra.SetPartitionsPk_k(k)

Bases: SetPartitionsAk_k

cardinality()

class sage.combinat.partition_algebra.SetPartitionsPkhalf_k(k)

Bases: SetPartitionsAkhalf_k

cardinality()

sage.combinat.partition_algebra.SetPartitionsRk(k)

Return the combinatorial class of set partitions of type \(R_k\).

EXAMPLES:

sage: SetPartitionsRk(3)
Set partitions of {1, ..., 3, -1, ..., -3} with at most 1 positive
 and negative entry in each block

class sage.combinat.partition_algebra.SetPartitionsRk_k(k)

Bases: SetPartitionsAk_k

cardinality()

class sage.combinat.partition_algebra.SetPartitionsRkhalf_k(k)

Bases: SetPartitionsAkhalf_k

cardinality()

sage.combinat.partition_algebra.SetPartitionsSk(k)

Return the combinatorial class of set partitions of type \(S_k\).

There is a bijection between these set partitions and the permutations of \(1, \ldots, k\).

EXAMPLES:

sage: S3 = SetPartitionsSk(3); S3
Set partitions of {1, ..., 3, -1, ..., -3} with propagating number 3
sage: S3.cardinality()
6

sage: S3.list()  #random
[{{2, -2}, {3, -3}, {1, -1}},
 {{1, -1}, {2, -3}, {3, -2}},
 {{2, -1}, {3, -3}, {1, -2}},
 {{1, -2}, {2, -3}, {3, -1}},
 {{1, -3}, {2, -1}, {3, -2}},
 {{1, -3}, {2, -2}, {3, -1}}]
sage: S3.first() #random
{{2, -2}, {3, -3}, {1, -1}}
sage: S3.last() #random
{{1, -3}, {2, -2}, {3, -1}}
sage: S3.random_element() #random
{{1, -3}, {2, -1}, {3, -2}}

sage: S3p5 = SetPartitionsSk(3.5); S3p5
Set partitions of {1, ..., 4, -1, ..., -4} with 4 and -4 in the same block and propagating number 4
sage: S3p5.cardinality()
6

sage: S3p5.list() #random
[{{2, -2}, {3, -3}, {1, -1}, {4, -4}},
 {{2, -3}, {1, -1}, {4, -4}, {3, -2}},
 {{2, -1}, {3, -3}, {1, -2}, {4, -4}},
 {{2, -3}, {1, -2}, {4, -4}, {3, -1}},
 {{1, -3}, {2, -1}, {4, -4}, {3, -2}},
 {{1, -3}, {2, -2}, {4, -4}, {3, -1}}]
sage: S3p5.first() #random
{{2, -2}, {3, -3}, {1, -1}, {4, -4}}
sage: S3p5.last() #random
{{1, -3}, {2, -2}, {4, -4}, {3, -1}}
sage: S3p5.random_element() #random
{{1, -3}, {2, -2}, {4, -4}, {3, -1}}

class sage.combinat.partition_algebra.SetPartitionsSk_k(k)

Bases: SetPartitionsAk_k

cardinality()

Return k!.

class sage.combinat.partition_algebra.SetPartitionsSkhalf_k(k)

Bases: SetPartitionsAkhalf_k

cardinality()

sage.combinat.partition_algebra.SetPartitionsTk(k)

Return the combinatorial class of set partitions of type \(T_k\).

These are planar set partitions where every block is of size 2.

EXAMPLES:

sage: T3 = SetPartitionsTk(3); T3
Set partitions of {1, ..., 3, -1, ..., -3} with block size 2 and that are planar
sage: T3.cardinality()
5

sage: T3.first() #random
{{1, -3}, {2, 3}, {-1, -2}}
sage: T3.last() #random
{{1, 2}, {3, -1}, {-3, -2}}
sage: T3.random_element() #random
{{1, -3}, {2, 3}, {-1, -2}}

sage: T2p5 = SetPartitionsTk(2.5); T2p5
Set partitions of {1, ..., 3, -1, ..., -3} with 3 and -3 in the same block and with block size 2 and that are planar
sage: T2p5.cardinality()
2

sage: T2p5.first() #random
{{2, -2}, {3, -3}, {1, -1}}
sage: T2p5.last() #random
{{1, 2}, {3, -3}, {-1, -2}}

class sage.combinat.partition_algebra.SetPartitionsTk_k(k)

Bases: SetPartitionsBk_k

cardinality()

class sage.combinat.partition_algebra.SetPartitionsTkhalf_k(k)

Bases: SetPartitionsBkhalf_k

cardinality()

class sage.combinat.partition_algebra.SetPartitionsXkElement(parent, s, check=True)

Bases: SetPartition

An element for the classes of SetPartitionXk where X is some letter.

check()

Check to make sure this is a set partition.

EXAMPLES:

sage: A2p5 = SetPartitionsAk(2.5)
sage: x = A2p5.first(); x
{{-3, -2, -1, 1, 2, 3}}
sage: x.check()
sage: y = A2p5.next(x); y
{{-3, 3}, {-2, -1, 1, 2}}
sage: y.check()

sage.combinat.partition_algebra.identity(k)

Return the identity set partition 1, -1, …, k, -k

EXAMPLES:

sage: import sage.combinat.partition_algebra as pa
sage: pa.identity(2)
{{2, -2}, {1, -1}}

sage.combinat.partition_algebra.is_planar(sp)

Return True if the diagram corresponding to the set partition is planar; otherwise, it returns False.

EXAMPLES:

sage: import sage.combinat.partition_algebra as pa
sage: pa.is_planar( pa.to_set_partition([[1,-2],[2,-1]]))
False
sage: pa.is_planar( pa.to_set_partition([[1,-1],[2,-2]]))
True

sage.combinat.partition_algebra.pair_to_graph(sp1, sp2)

Return a graph consisting of the disjoint union of the graphs of set partitions sp1 and sp2 along with edges joining the bottom row (negative numbers) of sp1 to the top row (positive numbers) of sp2.

The vertices of the graph sp1 appear in the result as pairs (k, 1), whereas the vertices of the graph sp2 appear as pairs (k, 2).

EXAMPLES:

sage: import sage.combinat.partition_algebra as pa
sage: sp1 = pa.to_set_partition([[1,-2],[2,-1]])
sage: sp2 = pa.to_set_partition([[1,-2],[2,-1]])
sage: g = pa.pair_to_graph( sp1, sp2 ); g
Graph on 8 vertices

sage: g.vertices(sort=False) #random
[(1, 2), (-1, 1), (-2, 2), (-1, 2), (-2, 1), (2, 1), (2, 2), (1, 1)]
sage: g.edges(sort=False) #random
[((1, 2), (-1, 1), None),
 ((1, 2), (-2, 2), None),
 ((-1, 1), (2, 1), None),
 ((-1, 2), (2, 2), None),
 ((-2, 1), (1, 1), None),
 ((-2, 1), (2, 2), None)]

Another example which used to be wrong until trac ticket #15958:

sage: sp3 = pa.to_set_partition([[1, -1], [2], [-2]])
sage: sp4 = pa.to_set_partition([[1], [-1], [2], [-2]])
sage: g = pa.pair_to_graph( sp3, sp4 ); g
Graph on 8 vertices

sage: g.vertices(sort=True)
[(-2, 1), (-2, 2), (-1, 1), (-1, 2), (1, 1), (1, 2), (2, 1), (2, 2)]
sage: g.edges(sort=True)
[((-2, 1), (2, 2), None), ((-1, 1), (1, 1), None),
 ((-1, 1), (1, 2), None)]

sage.combinat.partition_algebra.propagating_number(sp)

Return the propagating number of the set partition sp.

The propagating number is the number of blocks with both a positive and negative number.

EXAMPLES:

sage: import sage.combinat.partition_algebra as pa
sage: sp1 = pa.to_set_partition([[1,-2],[2,-1]])
sage: sp2 = pa.to_set_partition([[1,2],[-2,-1]])
sage: pa.propagating_number(sp1)
2
sage: pa.propagating_number(sp2)
0

sage.combinat.partition_algebra.set_partition_composition(sp1, sp2)

Return a tuple consisting of the composition of the set partitions sp1 and sp2 and the number of components removed from the middle rows of the graph.

EXAMPLES:

sage: import sage.combinat.partition_algebra as pa
sage: sp1 = pa.to_set_partition([[1,-2],[2,-1]])
sage: sp2 = pa.to_set_partition([[1,-2],[2,-1]])
sage: pa.set_partition_composition(sp1, sp2) == (pa.identity(2), 0)
True

sage.combinat.partition_algebra.to_graph(sp)

Return a graph representing the set partition sp.

EXAMPLES:

sage: import sage.combinat.partition_algebra as pa
sage: g = pa.to_graph( pa.to_set_partition([[1,-2],[2,-1]])); g
Graph on 4 vertices

sage: g.vertices(sort=False) #random
[1, 2, -2, -1]
sage: g.edges(sort=False) #random
[(1, -2, None), (2, -1, None)]

sage.combinat.partition_algebra.to_set_partition(l, k=None)

Convert a list of a list of numbers to a set partitions.

Each list of numbers in the outer list specifies the numbers contained in one of the blocks in the set partition.

If k is specified, then the set partition will be a set partition of 1, …, k, -1, …, -k. Otherwise, k will default to the minimum number needed to contain all of the specified numbers.

EXAMPLES:

sage: import sage.combinat.partition_algebra as pa
sage: pa.to_set_partition([[1,-1],[2,-2]]) == pa.identity(2)
True