The extremals of the Kahn-Saks inequality

TL;DR

This paper characterizes the extremal structures of log-concave sequences arising from the Kahn-Saks inequality in posets, revealing new extremals and connecting combinatorics with geometric principles.

Contribution

It provides a complete combinatorial characterization of the extremals of the Kahn-Saks inequality, linking log-concavity to geometric progressions and proving a partial conjecture.

Findings

Identified all possible extremal sequences in the Kahn-Saks inequality

Discovered new extremals not previously conjectured

Connected combinatorial extremals with geometric principles

Abstract

A classical result of Kahn and Saks states that given any partially ordered set with two distinguished elements, the number of linear extensions in which the ranks of the distinguished elements differ by $k$ is log-concave as a function of $k$. The log-concave sequences that can arise in this manner prove to exhibit a much richer structure, however, than is evident from log-concavity alone. The main result of this paper is a complete characterization of the extremals of the Kahn-Saks inequality: we obtain a detailed combinatorial understanding of where and what kind of geometric progressions can appear in these log-concave sequences. This settles a partial conjecture of Chan-Pak-Panova, while the analysis uncovers new extremals that were not previously conjectured. The proof relies on a much more general geometric mechanism -- a hard Lefschetz theorem for nef classes that was obtained in the setting of convex polytopes by Shenfeld and Van Handel -- which forms a model for the investigation of such structures in other combinatorial problems.