youngSubposet -- generates a subposet of Young's lattice

Synopsis

Usage:
P = youngSubposet(lo, hi)
P = youngSubposet hi
P = youngSubposet n
Inputs:
- lo, a list, which represents a partition
- hi, a list, which represents a partition
- n, an integer, the maximum size of a partition
Outputs:
- P, an object of class Poset

Description

Young’s lattice is the infinite lattice of all partitions with partial ordering given by componentwise linear ordering.

If n is specified, then the poset returned is the subposet of Young’s lattice given by the induced subposet of all partitions of size at most n.

i1 : youngSubposet 4

o1 = Poset{cache => CacheTable{...8...}                                                                                                                                                                                                                                }
           GroundSet => {{}, {1}, {2}, {1, 1}, {3}, {2, 1}, {1, 1, 1}, {4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}}
           RelationMatrix => | 1 1 1 1 1 1 1 1 1 1 1 1 |
                             | 0 1 1 1 1 1 1 1 1 1 1 1 |
                             | 0 0 1 0 1 1 0 1 1 1 1 0 |
                             | 0 0 0 1 0 1 1 0 1 1 1 1 |
                             | 0 0 0 0 1 0 0 1 1 0 0 0 |
                             | 0 0 0 0 0 1 0 0 1 1 1 0 |
                             | 0 0 0 0 0 0 1 0 0 0 1 1 |
                             | 0 0 0 0 0 0 0 1 0 0 0 0 |
                             | 0 0 0 0 0 0 0 0 1 0 0 0 |
                             | 0 0 0 0 0 0 0 0 0 1 0 0 |
                             | 0 0 0 0 0 0 0 0 0 0 1 0 |
                             | 0 0 0 0 0 0 0 0 0 0 0 1 |
           Relations => {{{}, {1}}, {{1}, {2}}, {{1}, {1, 1}}, {{2}, {3}}, {{2}, {2, 1}}, {{1, 1}, {2, 1}}, {{1, 1}, {1, 1, 1}}, {{3}, {4}}, {{3}, {3, 1}}, {{2, 1}, {3, 1}}, {{2, 1}, {2, 2}}, {{2, 1}, {2, 1, 1}}, {{1, 1, 1}, {2, 1, 1}}, {{1, 1, 1}, {1, 1, 1, 1}}}

o1 : Poset

If an upper partition, hi, is specified, then the returned poset is the closedInterval of the Young’s lattice between lo and hi, where lo either is the empty partition or is specified.

i2 : youngSubposet({3,1}, {4,2,1})

o2 = Poset{cache => CacheTable{}                                                                                                                                                                                                                                                                                                                                                                                                       }
           GroundSet => {{3, 1}, {3, 1, 1}, {3, 2}, {3, 2, 1}, {4, 1}, {4, 1, 1}, {4, 2}, {4, 2, 1}}
           RelationMatrix => | 1 1 1 1 1 1 1 1 |
                             | 0 1 0 1 0 1 0 1 |
                             | 0 0 1 1 0 0 1 1 |
                             | 0 0 0 1 0 0 0 1 |
                             | 0 0 0 0 1 1 1 1 |
                             | 0 0 0 0 0 1 0 1 |
                             | 0 0 0 0 0 0 1 1 |
                             | 0 0 0 0 0 0 0 1 |
           Relations => {{{3, 1}, {3, 1, 1}}, {{3, 1}, {3, 2}}, {{3, 1}, {3, 2, 1}}, {{3, 1}, {4, 1}}, {{3, 1}, {4, 1, 1}}, {{3, 1}, {4, 2}}, {{3, 1}, {4, 2, 1}}, {{3, 1, 1}, {3, 2, 1}}, {{3, 1, 1}, {4, 1, 1}}, {{3, 1, 1}, {4, 2, 1}}, {{3, 2}, {3, 2, 1}}, {{3, 2}, {4, 2}}, {{3, 2}, {4, 2, 1}}, {{3, 2, 1}, {4, 2, 1}}, {{4, 1}, {4, 1, 1}}, {{4, 1}, {4, 2}}, {{4, 1}, {4, 2, 1}}, {{4, 1, 1}, {4, 2, 1}}, {{4, 2}, {4, 2, 1}}}

o2 : Poset

Ways to use `youngSubposet` :

youngSubposet(List)
youngSubposet(List,List)
youngSubposet(ZZ)

youngSubposet -- generates a subposet of Young's lattice

Synopsis

Description

See also

Ways to use youngSubposet :

Ways to use `youngSubposet` :