On a conjecture about enumerating $(2+2)$-free posets

TL;DR

This paper provides a combinatorial proof for a conjecture related to counting unlabeled $(2+2)$-free posets based on their size and minimal elements, advancing understanding in poset enumeration.

Contribution

The paper offers the first combinatorial proof of a conjecture about the generating function for unlabeled $(2+2)$-free posets.

Findings

Confirmed the conjecture through combinatorial methods

Derived explicit generating functions for the posets

Enhanced understanding of poset enumeration techniques

Abstract

Recently, Kitaev and Remmel posed a conjecture concerning the generating function for the number of unlabeled $(2+2)$-free posets with respect to number of elements and number of minimal elements. In this paper, we present a combinatorial proof of this conjecture.