Variance vs. range for linear extensions, and balancing extensions in posets of bounded width

TL;DR

This paper explores the relationship between the structure of posets and the balancing of elements in their linear extensions, proving new implications for the Kahn-Saks conjecture and related sorting probability results.

Contribution

It establishes that large variance in element positions implies the Kahn-Saks conjecture holds for posets with bounded width, connecting variance and balancing properties.

Findings

Large variance implies the Kahn-Saks conjecture for bounded width posets.

Posets with large incomparability sets have large variance in linear extensions.

The results provide a new proof for sorting probabilities in Young diagrams.

Abstract

An old conjecture of Kahn and Saks says, roughly, that any poset $P$ of large enough width contains elements $x,y$ which are "balanced" in the sense that the probability that $x$ precedes $y$ in a uniformly random linear extension of $P$ is close to $1/2$. We show this implies the seemingly stronger statement that the same conclusion holds if, instead of large width, we assume only that, for some $x$, the number, $\pi(x)$, of elements of $P$ incomparable to $x$ is large. The implication follows from our two main results: first, that if $\pi(P):=\max \pi(x)$ is large then $P$ has large variance, i.e. there is a $y$ whose position in a uniform extension of $P$ has large variance; and second, that the conclusion of the Kahn-Saks Conjecture holds for $P$ with large variance and bounded width. These two assertions also yield an easy proof of a (not easy) result of Chan, Pak and Panova on "sorting probabilities" for Young diagrams, together with its natural generalization to higher dimensions.