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 \[\delta(P) \, := \, \min_{ x,y\in X} \, \bigl| \mathbf{ P} \, [L(x)\leq L(y) ] \ - \ \mathbf{ P} \, [L(y)\leq L(x) ] \bigr|\,, \] 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.
Categories: math.CO math.PR