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Posets :: dominanceLattice

dominanceLattice -- generates the dominance lattice of partitions of $n$

Synopsis

Description

The dominance lattice of partitons of n is the lattice of partitons of n under the dominance ordering. Suppose p and q are two partitions of n. Then p is less than or equal to q if and only if the k-th partial sum of p is at most the k-th partial sum of q, where the partitions are extended with zeros, as needed.
i1 : D = dominanceLattice 6;
i2 : closedInterval(D, {2,2,1,1}, {4,2})

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

o2 : Poset
For n ≤5, the dominance lattice of n is isomorphic to an appropriately long chain poset.
i3 : dominanceLattice 2 == chain 2

o3 = true
i4 : dominanceLattice 3 == chain 3

o4 = true
i5 : dominanceLattice 4 == chain 5

o5 = true
i6 : dominanceLattice 5 == chain 7

o6 = true

See also

Ways to use dominanceLattice :