The lattice of integer partitions

In this paper we study the lattice L_n of partitions of an integer n ordered by dominance. We show L_n to be isomorphic to an infimum subsemilattice under the component ordering of certain concave nondecreasing (n+1)-tuples. For L_n, we give the covering relation, maximal covering number, minimal chains, infimum and supremum irreducibles, a chain condition, distinguished intervals; and show that partition conjugation is a lattice antiautomorphism. L_n is shown to have no sublattice having five elements and rank two, and we characterize intervals generated by two cocovers. The Möbius function of L_n is computed and shown to be 0,1 or -1. We then give methods for studying classes of (0,1)-matrices with prescribed row and column sums and compute a lower bound for their cardinalities.