Self-dual interval orders and row-Fishburn matrices

TL;DR

This paper provides a combinatorial bijective proof linking self-dual interval orders and row-Fishburn matrices, confirming and refining previous generating function results.

Contribution

It introduces a bijection between matrix variations, offering a combinatorial proof of the relation and its refinement, advancing understanding of these combinatorial structures.

Findings

Established a bijection between matrix classes

Provided a combinatorial proof of the relation

Refined the relation between self-dual and row-Fishburn matrices

Abstract

Recently, Jel\'{i}nek derived that the number of self-dual interval orders of reduced size $n$ is twice the number of row-Fishburn matrices of size $n$ by using generating functions. In this paper, we present a bijective proof of this relation by establishing a bijection between two variations of upper-triangular matrices of nonnegative integers. Using the bijection, we provide a combinatorial proof of the refined relations between self-dual Fishburn matrices and row-Fishburn matrices in answer to a problem proposed by Jel\'{i}nek.