The lattice of integer partitions

AbstractIn this paper we study the lattice Ln of partitions of an integer n ordered by dominance. We show Ln to be isomorphic to an infimum subsemilattice under the component ordering of certain concave nondecreasing (n+1)-tuples. For Ln, 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. Ln is shown to have no sublattice having five elements and rank two, and we characterize intervals generated by two cocovers. The Möbius function of Ln 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