Super Partitions¶
AUTHORS:
- Mike Zabrocki
A super partition of size \(n\) and fermionic sector \(m\) is a pair consisting of a strict partition of some integer \(r\) of length \(m\) (that may end in a \(0\)) and an integer partition of \(n - r\).
This module provides tools for manipulating super partitions.
Super partitions are the indexing set for symmetric functions in super space.
Super partitions may be input in two different formats: one as a pair consisting of fermionic (strict partition) and a bosonic (partition) part and the other as a list of integer values where the negative entries come first and are listed in strict order followed by the positive values in weak order.
A super partition is displayed as two partitions separated by a semicolon as a default. Super partitions may also be displayed as a weakly increasing sequence of integers that are strict if the numbers are not positive.
These combinatorial objects index the space of symmetric polynomials in two sets of variables, one commuting and one anti-commuting, and they are known as symmetric functions in super space (hence the origin of the name super partitions).
EXAMPLES:
sage: SuperPartitions()
Super Partitions
sage: SuperPartitions(2)
Super Partitions of 2
sage: SuperPartitions(2).cardinality()
8
sage: SuperPartitions(4,2)
Super Partitions of 4 and of fermionic sector 2
sage: [[2,0],[1,1]] in SuperPartitions(4,2)
True
sage: [[1,0],[1,1]] in SuperPartitions(4,2)
False
sage: [[1,0],[2,1]] in SuperPartitions(4)
True
sage: [[1,0],[2,2,1]] in SuperPartitions(4)
False
sage: [[1,0],[2,1]] in SuperPartitions()
True
sage: [[1,1],[2,1]] in SuperPartitions()
False
sage: [-2, 0, 1, 1] in SuperPartitions(4,2)
True
sage: [-1, 0, 1, 1] in SuperPartitions(4,2)
False
sage: [-2, -2, 2, 1] in SuperPartitions(7,2)
False
REFERENCES:
-
sage.combinat.superpartition.
SuperPartition
¶ A super partition.
A super partition of size \(n\) and fermionic sector \(m\) is a pair consisting of a strict partition of some integer \(r\) of length \(m\) (that may end in a \(0\)) and an integer partition of \(n - r\).
EXAMPLES:
sage: sp = SuperPartition([[1,0],[2,2,1]]); sp [1, 0; 2, 2, 1] sage: sp[0] (1, 0) sage: sp[1] (2, 2, 1) sage: sp.fermionic_degree() 2 sage: sp.bosonic_degree() 6 sage: sp.length() 5 sage: sp.conjugate() [4, 2; ]
-
sage.combinat.superpartition.
SuperPartitions
¶ Super partitions.
A super partition of size \(n\) and fermionic sector \(m\) is a pair consisting of a strict partition of some integer \(r\) of length \(m\) (that may end in a \(0\)) and an integer partition of \(n - r\).
INPUT:
n
– an integer (optional: defaultNone
)m
– ifn
is specified, an integer (optional: defaultNone
)
Super partitions are the indexing set for symmetric functions in super space.
EXAMPLES:
sage: SuperPartitions() Super Partitions sage: SuperPartitions(2) Super Partitions of 2 sage: SuperPartitions(2).cardinality() 8 sage: SuperPartitions(4,2) Super Partitions of 4 and of fermionic sector 2 sage: [[2,0],[1,1]] in SuperPartitions(4,2) True sage: [[1,0],[1,1]] in SuperPartitions(4,2) False sage: [[1,0],[2,1]] in SuperPartitions(4) True sage: [[1,0],[2,2,1]] in SuperPartitions(4) False sage: [[1,0],[2,1]] in SuperPartitions() True sage: [[1,1],[2,1]] in SuperPartitions() False
-
class
sage.combinat.superpartition.
SuperPartitions_all
¶ Bases:
sage.combinat.superpartition.SuperPartitions
Initialize
self
.
-
class
sage.combinat.superpartition.
SuperPartitions_n
(n)¶ Bases:
sage.combinat.superpartition.SuperPartitions
Initialize
self
.
-
class
sage.combinat.superpartition.
SuperPartitions_n_m
(n, m)¶ Bases:
sage.combinat.superpartition.SuperPartitions
Initialize
self
.