Counting proper mergings of chains and antichains

A proper merging of two disjoint quasi-ordered sets (P,←P)(P,←P) and (Q,←Q)(Q,←Q), denoted by PP and QQ, respectively, is a quasi-order on the union of PP and QQ such that the restriction to PP or QQ yields the original quasi-orders on those sets and such that no elements of PP and QQ are identified. In this article, we consider the cases where PP and QQ are chains, where PP and QQ are antichains, and where PP is an antichain and QQ is a chain. We give formulas that determine the number of proper mergings in all three cases. We also introduce two new bijections from proper mergings of two chains to plane partitions and from proper mergings of an antichain and a chain to monotone colorings of complete bipartite digraphs.