A proper merging of two disjoint quasi-ordered sets P and Q is a
quasi-order on the union of P and Q such that the restriction to P and
Q yields the original quasi-order again and such that no elements of P and
Q are identified. In this article, we consider the cases where P and Q
are chains, where P and Q are antichains, and where P is an antichain and
Q is a chain. We give formulas that determine the number of proper mergings
in all three cases, and 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. Additionally, we
use these bijections to count the Galois connections between two chains, and
between a chain and a Boolean lattice respectively.Comment: 36 pages, 15 figures, 5 table