The cross-product conjecture for width two posets

TL;DR

This paper proves the cross-product conjecture for width-two posets using algebraic and combinatorial methods, extending it to a four-parameter family and a q-analogue, and explores related inequalities.

Contribution

It provides two novel proofs of the CPC for width-two posets, generalizing it and connecting it to other poset inequalities.

Findings

Algebraic proof generalizes CPC to four parameters

Combinatorial proof extends CPC to a q-analogue

Establishes relationships between CPC and other inequalities

Abstract

The cross--product conjecture (CPC) of Brightwell, Felsner and Trotter (1995) is a two-parameter quadratic inequality for the number of linear extensions of a poset $P= (X, \prec)$ with given value differences on three distinct elements in $X$. We give two different proofs of this inequality for posets of width two. The first proof is algebraic and generalizes CPC to a four-parameter family. The second proof is combinatorial and extends CPC to a $q$-analogue. Further applications include relationships between CPC and other poset inequalities, including a new $q$-analogue of the Kahn--Saks inequality.