Sorting probability for large Young diagrams

Abstract

For a finite poset $P=(X,\prec )$, let ${\mathcal{L}}_P$ denote the set of linear extensions of $P$. The sorting probability $\delta (P)$ is defined as \[δ(P) \, := \, \min_{x,y\in X} \, \bigl| \mathbf{P} \, [L(x)\leq L(y) ] \ - \ \mathbf{P} \, [L(y)\leq L(x) ] \bigr|\,, \] where $L\in {\mathcal{L}}_P$ is a uniform linear extension of $P$. We give asymptotic upper bounds on sorting probabilities for posets associated with large Young diagrams and large skew Young diagrams, with bounded number of rows.

57 pages, 5 pages