Alternative Approaches for Counting Weakly Increasing Matrices

TL;DR

This paper introduces two novel combinatorial methods for counting two-row weakly increasing matrices, linking them to Kekulé structures and non-intersecting lattice paths, thus revealing interdisciplinary connections.

Contribution

It proposes two new approaches for counting weakly increasing matrices, establishing bijections and reductions to other combinatorial problems.

Findings

Established a bijection with Kekulé structures for certain hexagonal benzenoids.

Reduced the counting problem to pairs of non-intersecting lattice paths.

Revealed connections between matrix counting and chemical graph theory.

Abstract

This paper presents two alternative approaches for counting the number of two-row weakly increasing matrices, which are $2\times n$ matrices whose entries are integers from $1$ to $k$ and are weakly increasing along all rows and columns, for any positive integers $n$ and $k$. The first approach establishes a bijection between the set of such matrices and the set of Kekul\'e structures for certain hexagonal benzenoids. The second approach reduces the problem to counting the number of pairs of non-intersecting lattice paths. These approaches reveal interesting connections between combinatorial problems that arise in different domains.