Chain Partitions of Products of Two Chains

We characterize sequences of positive integers (m1, …, mp),m1⩾…⩾mp>0, for which the productn×k(ofn-element andk-element chains),n⩾k, can be partitioned into chainsC1, …, Cpsuch that |Ci|=mi, fori=1, …, p. The main result says that such a partition exists if and only if ∑ji=1 mi⩽∑ji=1 (k+n+1−2i), forj=1, …, kand ∑pi=1 mi mi=nk.