The Generalized Interval Polynomial of a Poset

TL;DR

This paper introduces a generating polynomial for finite posets that counts specific subsets, explores its properties under poset operations, and demonstrates its ability to distinguish certain poset classes, with implications for quantum gravity models.

Contribution

It defines a new polynomial invariant for finite posets, analyzes its behavior under operations, and shows it uniquely identifies series-parallel posets, linking combinatorics to quantum gravity.

Findings

The polynomial counts upsets, downsets, and their intersections.

It distinguishes series-parallel posets from others.

Connections to the causal set approach to quantum gravity are discussed.

Abstract

For any finite poset we define a generating polynomial counting upsets, downsets, and their intersection. We investigate the behaviour of this polynomial with respect to poset operations, show that it distinguishes series-parallel posets, and comment on connections to the causal set approach to quantum gravity.