Variance vs. range for linear extensions, and balancing extensions in posets of bounded width
Max Aires, Jeff Kahn
TL;DR
The paper investigates balancing in linear extensions of posets by linking incomparability, width, and the variance of element positions. It proves that large incomparability counts $\pi(P)$ force large variance (via a global-to-local variance principle) and that large variance combined with bounded width yields near-1/2 balancing probabilities, effectively a bounded-width version of the Kahn–Saks conjecture. The main technical advances include a width-bounded variance bound (Theorem Twsig), a reduction from large $\pi$ to large average variance (Theorem pi-var and its unidirectional specialization), and a width-2 poset main lemma built on Shepp, Stanley, and Grünbaum machinery. These results unify several balancing questions and extend sorting-probability results for Young diagrams to higher dimensions, offering new equivalences and consequences for poset sorting phenomena.
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, $π(x)$, of elements of $P$ incomparable to $x$ is large. The implication follows from our two main results: first, that if $π(P):=\max π(x)$ 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.