Counting Self-Dual Interval Orders

TL;DR

This paper introduces a new method for deriving generating functions for interval orders, including self-dual ones, and establishes a bijection between self-dual interval orders and certain upper-triangular matrices.

Contribution

It generalizes previous enumeration results and provides a novel enumeration and bijective characterization of self-dual interval orders.

Findings

Derived formulas for generating functions of interval orders

Enumerated self-dual interval orders with respect to key statistics

Established a bijection between self-dual interval orders and upper-triangular matrices

Abstract

In this paper, we present a new method to derive formulas for the generating functions of interval orders, counted with respect to their size, magnitude, and number of minimal and maximal elements. Our method allows us not only to generalize previous results on refined enumeration of general interval orders, but also to enumerate self-dual interval orders with respect to analogous statistics. Using the newly derived generating function formulas, we are able to prove a bijective relationship between self-dual interval orders and upper-triangular matrices with no zero rows. Previously, a similar bijective relationship has been established between general interval orders and upper-triangular matrices with no zero rows and columns.