Garland Recurrences

TL;DR

This paper introduces recurrence relations for counting linear extensions of garland posets, generalizing fence posets, and demonstrates their divergence and computational implementation, with results available on OEIS.

Contribution

It provides new recurrence relations for garland posets and applies them to analyze their generating series and computational methods.

Findings

Recurrence relations for linear extensions of garland posets.

Divergence of the generating series for these posets.

Efficient Python implementation for computing sequence terms.

Abstract

Partially ordered sets have received much attention in recent years, not just due to their usefulness in combinatorics and abstract algebra, but also due to their practical applications in fields ranging from chemistry to macroeconomics. The garland or double fence $G_N$ is a partially ordered set with $2N$ elements which generalizes the well-known fence or zigzag poset. The main result of this paper is recurrence relations for enumerating the linear extensions of $G_N$. These recurrences were then applied to prove divergence of the standard type $G_N$-generating series. When coded in Python, it provides a fast method for computing $e(G_n)$ for arbitrary $n$. The first 200 terms of this sequence are published online under OEIS A227656.