Posets of Hyper $(b,t)$-ary Partitions are Distributive Lattices

Abstract

Posets of integer partitions, ordered by refinement, were criticized for unruly structural behavior by both Birkhoff and Ziegler. However, recent work of Propp, McConville, and Sagan demonstrated that such posets of hyperbinary partitions are distributive, with posets of join irreducible elements isomorphic to the well-studied class of fence posets. We provide a generalization of this result for hyper $(b,t)$-ary partitions, where all parts are powers of some positive integer $b$, and multiplicities are restricted to $t\ge b$. Then, we show that posets of hyper $(b,t)$-ary partitions are indeed also distributive lattices.