Chain Partitions of Products of Two Chains

摘要

We characterize sequences of positive integers (m_1, ..., m_p), m_1 >= ... >= m_p > 0, for which the product n * k (of n-element and k-element chains), n >= k, can be partitioned into chains C_1, ..., C_p such that |C_i| = m_i, for i = 1, ..., p. The main result says that such a partition exists if and only if #SIGMA#_(i = 1)~i m_i <= #SIGMA#_(i=1)~i (k + n + 1 2i), for i = 1, ..., k and #SIGMA#_(i = 1)~p m_i = nk.