Proper Mergings of Stars and Chains are Counted by Sums of Antidiagonals in Certain Convolution Arrays -- The Details

Abstract

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 $P$ or $Q$ yields the original quasi-order again and such that no elements of $P$ and $Q$ are identified. In this article, we determine the number of proper mergings in the case where $P$ is a star (i.e. an antichain with a smallest element adjoined), and $Q$ is a chain. We show that the lattice of proper mergings of an $m$-antichain and an $n$-chain, previously investigated by the author, is a quotient lattice of the lattice of proper mergings of an $m$-star and an $n$-chain, and we determine the number of proper mergings of an $m$-star and an $n$-chain by counting the number of congruence classes and by determining their cardinalities. Additionally, we compute the number of Galois connections between certain modified Boolean lattices and chains.