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

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 $1\mathrm{/}2$. 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.