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

## Synopsis

• Usage:
P = dominanceLattice n
• Inputs:
• Outputs:
• P, an instance of the type Poset,

## Description

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 $n \leq 5$, 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