Fertility Monotonicity and Average Complexity of the Stack-Sorting Map

Abstract

Let ${\mathcal{D}}_n$ denote the average number of iterations of West's stack-sorting map $s$ that are needed to sort a permutation in $S_n$ into the identity permutation $123\cdots n$. We prove that \[0.62433\approxλ\leq\liminf_{n\to\infty}\frac{\mathcal D_n}{n}\leq\limsup_{n\to\infty}\frac{\mathcal D_n}{n}\leq \frac{3}{5}(7-8\log 2)\approx 0.87289,\] where $\hat{I}»$ is the Golomb-Dickman constant. Our lower bound improves upon West's lower bound of $0.23$, and our upper bound is the first improvement upon the trivial upper bound of $1$. We then show that fertilities of permutations increase monotonically upon iterations of $s$. More precisely, we prove that $|s^{-1}(\ddot{I}ƒ)|\le |s^{-1}(s(\ddot{I}ƒ))|$ for all $\ddot{I}ƒ\in S_n$, where equality holds if and only if $\ddot{I}ƒ=123\cdots n$. This is the first theorem that manifests a law-of-diminishing-returns philosophy for the stack-sorting map that Bóna has proposed. Along the way, we note some connections between the stack-sorting map and the right and left weak orders on $S_n$.

17 pages, 4 figures