Partition/Diagram Algebras

class sage.combinat.partition_algebra.PartitionAlgebraElement_ak(M, x)

Bases: sage.combinat.partition_algebra.PartitionAlgebraElement_generic

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

class sage.combinat.partition_algebra.PartitionAlgebraElement_bk(M, x)

Bases: sage.combinat.partition_algebra.PartitionAlgebraElement_generic

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

class sage.combinat.partition_algebra.PartitionAlgebraElement_generic(M, x)

Bases: sage.combinat.free_module.CombinatorialFreeModuleElement

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

class sage.combinat.partition_algebra.PartitionAlgebraElement_pk(M, x)

Bases: sage.combinat.partition_algebra.PartitionAlgebraElement_generic

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

class sage.combinat.partition_algebra.PartitionAlgebraElement_prk(M, x)

Bases: sage.combinat.partition_algebra.PartitionAlgebraElement_generic

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

class sage.combinat.partition_algebra.PartitionAlgebraElement_rk(M, x)

Bases: sage.combinat.partition_algebra.PartitionAlgebraElement_generic

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

class sage.combinat.partition_algebra.PartitionAlgebraElement_sk(M, x)

Bases: sage.combinat.partition_algebra.PartitionAlgebraElement_generic

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

class sage.combinat.partition_algebra.PartitionAlgebraElement_tk(M, x)

Bases: sage.combinat.partition_algebra.PartitionAlgebraElement_generic

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

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

Bases: sage.combinat.partition_algebra.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: sage.combinat.partition_algebra.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: sage.combinat.combinatorial_algebra.CombinatorialAlgebra

EXAMPLES:

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

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

Bases: sage.combinat.partition_algebra.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: sage.combinat.partition_algebra.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: sage.combinat.partition_algebra.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: sage.combinat.partition_algebra.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: sage.combinat.partition_algebra.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)

Returns 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: sage.combinat.set_partition.SetPartitions_set

Element

alias of SetPartitionsXkElement

class sage.combinat.partition_algebra.SetPartitionsAkhalf_k(k)

Bases: sage.combinat.set_partition.SetPartitions_set

Element

alias of SetPartitionsXkElement

sage.combinat.partition_algebra.SetPartitionsBk(k)

Returns 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: sage.combinat.partition_algebra.SetPartitionsAk_k

cardinality()

Returns 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: sage.combinat.partition_algebra.SetPartitionsAkhalf_k

cardinality()

sage.combinat.partition_algebra.SetPartitionsIk(k)

Returns 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}, ..., {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: sage.combinat.partition_algebra.SetPartitionsAk_k

cardinality()

class sage.combinat.partition_algebra.SetPartitionsIkhalf_k(k)

Bases: sage.combinat.partition_algebra.SetPartitionsAkhalf_k

cardinality()

sage.combinat.partition_algebra.SetPartitionsPRk(k)

class sage.combinat.partition_algebra.SetPartitionsPRk_k(k)

Bases: sage.combinat.partition_algebra.SetPartitionsRk_k

cardinality()

class sage.combinat.partition_algebra.SetPartitionsPRkhalf_k(k)

Bases: sage.combinat.partition_algebra.SetPartitionsRkhalf_k

cardinality()

sage.combinat.partition_algebra.SetPartitionsPk(k)

Returns 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: sage.combinat.partition_algebra.SetPartitionsAk_k

cardinality()

class sage.combinat.partition_algebra.SetPartitionsPkhalf_k(k)

Bases: sage.combinat.partition_algebra.SetPartitionsAkhalf_k

cardinality()

sage.combinat.partition_algebra.SetPartitionsRk(k)

class sage.combinat.partition_algebra.SetPartitionsRk_k(k)

Bases: sage.combinat.partition_algebra.SetPartitionsAk_k

cardinality()

class sage.combinat.partition_algebra.SetPartitionsRkhalf_k(k)

Bases: sage.combinat.partition_algebra.SetPartitionsAkhalf_k

cardinality()

sage.combinat.partition_algebra.SetPartitionsSk(k)

Returns the combinatorial class of set partitions of type S_k. There is a bijection between these set partitions and the permutations of 1, ..., 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: sage.combinat.partition_algebra.SetPartitionsAk_k

cardinality()

Returns k!.

class sage.combinat.partition_algebra.SetPartitionsSkhalf_k(k)

Bases: sage.combinat.partition_algebra.SetPartitionsAkhalf_k

cardinality()

sage: ks = [2.5, 3.5, 4.5, 5.5]
sage: sks = [SetPartitionsSk(k) for k in ks]
sage: all(sk.cardinality() == len(sk.list()) for sk in sks)
True

sage.combinat.partition_algebra.SetPartitionsTk(k)

Returns 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: sage.combinat.partition_algebra.SetPartitionsBk_k

cardinality()

class sage.combinat.partition_algebra.SetPartitionsTkhalf_k(k)

Bases: sage.combinat.partition_algebra.SetPartitionsBkhalf_k

cardinality()

class sage.combinat.partition_algebra.SetPartitionsXkElement(parent, s)

Bases: sage.combinat.set_partition.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 # random
{{1, 2, 3, -1, -3, -2}}
sage: x.check()
sage: y = A2p5.next(x); y
{{-3, -2, -1, 2, 3}, {1}}
sage: y.check()

sage.combinat.partition_algebra.create_set_partition_function(letter, k)

EXAMPLES:

sage: from sage.combinat.partition_algebra import create_set_partition_function
sage: create_set_partition_function('A', 3)
Set partitions of {1, ..., 3, -1, ..., -3}

sage.combinat.partition_algebra.identity(k)

Returns 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)

Returns 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() #random
[(1, 2), (-1, 1), (-2, 2), (-1, 2), (-2, 1), (2, 1), (2, 2), (1, 1)]
sage: g.edges() #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()
[(-2, 1), (-2, 2), (-1, 1), (-1, 2), (1, 1), (1, 2), (2, 1), (2, 2)]
sage: g.edges()
[((-2, 1), (2, 2), None), ((-1, 1), (1, 1), None),
 ((-1, 1), (1, 2), None)]

sage.combinat.partition_algebra.propagating_number(sp)

Returns 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)

Returns 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)

Returns 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() #random
[1, 2, -2, -1]
sage: g.edges() #random
[(1, -2, None), (2, -1, None)]

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

Coverts 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