Dominance order - AbsoluteAstronomy.com

Dominance order is a partial order on the set of partitions

Partition (number theory)

In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered to be the same partition; if order matters then the sum becomes a...

of a positive integer n that plays an important role in algebraic combinatorics

Algebraic combinatorics

Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra....

and representation theory

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

, especially in the context of symmetric function

Symmetric function

In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity...

s and representation theory of the symmetric group

Representation theory of the symmetric group

In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to problems of quantum...

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If p₁,p₂,… and q₁,q₂,… are partitions of n, with the parts arranged in the weakly decreasing order, then p precedes q in the dominance order if for any k ≥ 1, the sum of the k largest parts of p is less than or equal to the sum of the k largest parts of q:

if and only if

for all

In this definition, partitions are extended by appending zero parts at the end as necessary.

Among the partitions of n, (1,…,1) is the smallest and (n) is the largest.
The dominance ordering implies lexicographical ordering, i.e.