In discrete mathematics, dominance order (synonyms: dominance ordering, majorization order, natural ordering) is a partial order on the set of partitions of a positive integer n that plays an important role in algebraic combinatorics and representation theory, especially in the context of symmetric functions and representation theory of the symmetric group. Definition If p = (p1,p2,…) and q = (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: In this definition, partitions are extended by appending zero parts at the end as necessary. Properties of the dominance ordering Among the partitions of n, (1,…,1) is the smallest and (n) is the largest. The dominance ordering implies lexicographical ordering, i.e. if p dominates q and p ≠ q, then for the smallest i such that pi ≠ qi one has pi > qi. The poset of partitions of n is linearly ordered (and is equivalent to lexicographical ordering) if and only if n ≤ 5. It is graded if and only if n ≤ 6. See image at right for an example. A partition p covers a partition q if and only if pi = qi + 1, pk = qk − 1, pj = qj for all j ≠ i,k and either (1) k = i + 1 or (2) qi = qk (Brylawski, Prop. 2.3). Starting from the Young diagram of q, the Young diagram of p is obtained from it by first removing the last box of row k and then appending it either to the end of the immediately preceding row k − 1, or to the end of row i < k if the rows i through k of the Young diagram of q all have the same length. Every partition p has a conjugate (or dual) partition p′, whose Young diagram is the transpose of the Young diagram of p. This operation reverses the dominance ordering: if and only if The dominance ordering determines the inclusions between the Zariski closures of the conjugacy classes of nilpotent matrices. Lattice structure Partitions of n form a lattice under the dominance ordering, denoted Ln, and the operation of conjugation is an antiautomorphism of this lattice.