Partition/Diagram Algebras¶

class sage.combinat.partition_algebra.PartitionAlgebraElement_ak(M, x)¶: Bases: sage.combinat.partition_algebra.PartitionAlgebraElement_generic

class sage.combinat.partition_algebra.PartitionAlgebraElement_bk(M, x)¶: Bases: sage.combinat.partition_algebra.PartitionAlgebraElement_generic

class sage.combinat.partition_algebra.PartitionAlgebraElement_generic(M, x)¶: Bases: sage.combinat.free_module.CombinatorialFreeModuleElement

class sage.combinat.partition_algebra.PartitionAlgebraElement_pk(M, x)¶: Bases: sage.combinat.partition_algebra.PartitionAlgebraElement_generic

class sage.combinat.partition_algebra.PartitionAlgebraElement_prk(M, x)¶: Bases: sage.combinat.partition_algebra.PartitionAlgebraElement_generic

class sage.combinat.partition_algebra.PartitionAlgebraElement_rk(M, x)¶: Bases: sage.combinat.partition_algebra.PartitionAlgebraElement_generic

class sage.combinat.partition_algebra.PartitionAlgebraElement_sk(M, x)¶: Bases: sage.combinat.partition_algebra.PartitionAlgebraElement_generic

class sage.combinat.partition_algebra.PartitionAlgebraElement_tk(M, x)¶: Bases: sage.combinat.partition_algebra.PartitionAlgebraElement_generic

class sage.combinat.partition_algebra.PartitionAlgebra_ak(R, k, n, name=None)¶: Bases: sage.combinat.partition_algebra.PartitionAlgebra_generic

class sage.combinat.partition_algebra.PartitionAlgebra_bk(R, k, n, name=None)¶: Bases: sage.combinat.partition_algebra.PartitionAlgebra_generic

class sage.combinat.partition_algebra.PartitionAlgebra_generic(R, cclass, n, k, name=None, prefix=None)¶: Bases: sage.combinat.combinatorial_algebra.CombinatorialAlgebra

class sage.combinat.partition_algebra.PartitionAlgebra_pk(R, k, n, name=None)¶: Bases: sage.combinat.partition_algebra.PartitionAlgebra_generic

class sage.combinat.partition_algebra.PartitionAlgebra_prk(R, k, n, name=None)¶: Bases: sage.combinat.partition_algebra.PartitionAlgebra_generic

class sage.combinat.partition_algebra.PartitionAlgebra_rk(R, k, n, name=None)¶: Bases: sage.combinat.partition_algebra.PartitionAlgebra_generic

class sage.combinat.partition_algebra.PartitionAlgebra_sk(R, k, n, name=None)¶: Bases: sage.combinat.partition_algebra.PartitionAlgebra_generic

class sage.combinat.partition_algebra.PartitionAlgebra_tk(R, k, n, name=None)¶: Bases: sage.combinat.partition_algebra.PartitionAlgebra_generic

class sage.combinat.partition_algebra.SetPartitionsAk_k(k)¶: Bases: sage.combinat.set_partition.SetPartitions_set

class sage.combinat.partition_algebra.SetPartitionsAkhalf_k(k)¶

Bases: sage.combinat.combinat.CombinatorialClass

cardinality()¶

TESTS:

sage: SetPartitionsAk(1.5).cardinality()
5
sage: SetPartitionsAk(2.5).cardinality()
52
sage: SetPartitionsAk(3.5).cardinality()
877

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()¶

TESTS:

sage: B3p5 = SetPartitionsBk(3.5)
sage: B3p5.cardinality()
15

class sage.combinat.partition_algebra.SetPartitionsIk_k(k)¶

Bases: sage.combinat.partition_algebra.SetPartitionsAk_k

cardinality()¶

TESTS:

sage: SetPartitionsIk(2).cardinality()
13

class sage.combinat.partition_algebra.SetPartitionsIkhalf_k(k)¶

Bases: sage.combinat.partition_algebra.SetPartitionsAkhalf_k

cardinality()¶

TESTS:

sage: SetPartitionsIk(1.5).cardinality()
4
sage: SetPartitionsIk(2.5).cardinality()
50
sage: SetPartitionsIk(3.5).cardinality()
871

class sage.combinat.partition_algebra.SetPartitionsPRk_k(k)¶

Bases: sage.combinat.partition_algebra.SetPartitionsRk_k

cardinality()¶

TESTS:

sage: SetPartitionsPRk(2).cardinality()
6
sage: SetPartitionsPRk(3).cardinality()
20
sage: SetPartitionsPRk(4).cardinality()
70
sage: SetPartitionsPRk(5).cardinality()
252

class sage.combinat.partition_algebra.SetPartitionsPRkhalf_k(k)¶

Bases: sage.combinat.partition_algebra.SetPartitionsRkhalf_k

cardinality()¶

TESTS:

sage: SetPartitionsPRk(2.5).cardinality()
6
sage: SetPartitionsPRk(3.5).cardinality()
20
sage: SetPartitionsPRk(4.5).cardinality()
70

class sage.combinat.partition_algebra.SetPartitionsPk_k(k)¶

Bases: sage.combinat.partition_algebra.SetPartitionsAk_k

cardinality()¶

TESTS:

sage: SetPartitionsPk(2).cardinality()
14
sage: SetPartitionsPk(3).cardinality()
132
sage: SetPartitionsPk(4).cardinality()
1430

class sage.combinat.partition_algebra.SetPartitionsPkhalf_k(k)¶

Bases: sage.combinat.partition_algebra.SetPartitionsAkhalf_k

cardinality()¶

TESTS:

sage: SetPartitionsPk(2.5).cardinality()
42
sage: SetPartitionsPk(1.5).cardinality()
5

class sage.combinat.partition_algebra.SetPartitionsRk_k(k)¶

Bases: sage.combinat.partition_algebra.SetPartitionsAk_k

cardinality()¶

TESTS:

sage: SetPartitionsRk(2).cardinality()
7
sage: SetPartitionsRk(3).cardinality()
34
sage: SetPartitionsRk(4).cardinality()
209
sage: SetPartitionsRk(5).cardinality()
1546

class sage.combinat.partition_algebra.SetPartitionsRkhalf_k(k)¶

Bases: sage.combinat.partition_algebra.SetPartitionsAkhalf_k

cardinality()¶

TESTS:

sage: SetPartitionsRk(2.5).cardinality()
7
sage: SetPartitionsRk(3.5).cardinality()
34
sage: SetPartitionsRk(4.5).cardinality()
209

class sage.combinat.partition_algebra.SetPartitionsSk_k(k)¶

Bases: sage.combinat.partition_algebra.SetPartitionsAk_k

cardinality()¶

Returns k!.

TESTS:

sage: SetPartitionsSk(2).cardinality()
2
sage: SetPartitionsSk(3).cardinality()
6
sage: SetPartitionsSk(4).cardinality()
24
sage: SetPartitionsSk(5).cardinality()
120

class sage.combinat.partition_algebra.SetPartitionsSkhalf_k(k)¶

Bases: sage.combinat.partition_algebra.SetPartitionsAkhalf_k

cardinality()¶

TESTS:

sage: SetPartitionsSk(2.5).cardinality()
2
sage: SetPartitionsSk(3.5).cardinality()
6
sage: SetPartitionsSk(4.5).cardinality()
24

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

class sage.combinat.partition_algebra.SetPartitionsTk_k(k)¶

Bases: sage.combinat.partition_algebra.SetPartitionsBk_k

cardinality()¶

TESTS:

sage: SetPartitionsTk(2).cardinality()
2
sage: SetPartitionsTk(3).cardinality()
5
sage: SetPartitionsTk(4).cardinality()
14
sage: SetPartitionsTk(5).cardinality()
42

class sage.combinat.partition_algebra.SetPartitionsTkhalf_k(k)¶

Bases: sage.combinat.partition_algebra.SetPartitionsBkhalf_k

cardinality()¶

TESTS:

sage: SetPartitionsTk(2.5).cardinality()
2
sage: SetPartitionsTk(3.5).cardinality()
5
sage: SetPartitionsTk(4.5).cardinality()
14

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)¶

Returns a graph consisting 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.

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)]

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

Partition/Diagram Algebras¶

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