Decomposing labeled interval orders as pairs of permutations

TL;DR

This paper introduces ballot matrices and a bijection with labeled interval orders, providing a new combinatorial formula for their enumeration by decomposing them into permutation-inversion table pairs.

Contribution

It presents a novel combinatorial structure called ballot matrices and establishes a bijection with labeled interval orders, enabling enumeration through permutation-inversion table pairs.

Findings

Established a sign reversing involution on ballot matrices.

Proved a bijection between fixed points and labeled interval orders.

Derived a new enumeration formula for labeled interval orders.

Abstract

We introduce ballot matrices, a signed combinatorial structure whose definition naturally follows from the generating function for labeled interval orders. A sign reversing involution on ballot matrices is defined. We show that matrices fixed under this involution are in bijection with labeled interval orders and that they decompose to a pair consisting of a permutation and an inversion table. To fully classify such pairs, results pertaining to the enumeration of permutations having a given set of ascent bottoms are given. This allows for a new formula for the number of labeled interval orders.