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

TL;DR

This paper counts the number of proper mergings between a star and a chain in quasi-ordered sets, linking the enumeration to sums of antidiagonals in convolution arrays, and explores related Galois connections.

Contribution

It introduces a novel counting method for proper mergings of stars and chains using convolution array antidiagonals and analyzes their lattice structure.

Findings

Number of proper mergings expressed via sums of antidiagonals

Lattice of proper mergings is a quotient of a known lattice

Computed Galois connections between modified Boolean lattices and chains

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.