Extensions of the Kahn--Saks inequality for posets of width two

TL;DR

This paper provides a new proof of the Kahn--Saks inequality for width-two posets using lattice path injections, and extends it with a q-analogue, multivariate generalization, and equality conditions.

Contribution

It introduces a novel proof technique for the inequality and extends it with new variants and detailed equality condition analysis.

Findings

New proof of Kahn--Saks inequality for width-two posets

Development of a q-analogue and multivariate generalization

Analysis of equality conditions in the inequality

Abstract

The Kahn--Saks inequality is a classical result on the number of linear extensions of finite posets. We give a new proof of this inequality for posets of width two using explicit injections of lattice paths. As a consequence we obtain a $q$-analogue, a multivariate generalization and an equality condition in this case. We also discuss the equality conditions of the Kahn--Saks inequality for general posets and prove several implications between conditions conjectured to be equivalent.