The dominance lattice of partitions of n is the lattice of partitions 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 = Relation Matrix: | 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 | 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 |