Counting Proper Mergings of Chains and Antichains

TL;DR

This paper provides formulas for counting proper mergings of chains and antichains, introduces bijections to combinatorial objects, and applies these to count Galois connections in specific lattice structures.

Contribution

It offers new formulas for counting proper mergings of specific quasi-ordered sets and establishes bijections to combinatorial objects, enabling enumeration of Galois connections.

Findings

Formulas for proper mergings of chains and antichains

Bijections to plane partitions and monotone colorings

Counting Galois connections in specific lattice structures

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 $P$ 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.