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

Synopsis

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

See also

Ways to use dominanceLattice :

For the programmer

The object dominanceLattice is a method function.