Fertility monotonicity and average complexity of the stack-sorting map

Abstract

Let Dn denote the average number of iterations of West’s stack-sorting map s that are needed to sort a permutation in Sn into the identity permutation 123⋯n. We prove that 0.62433≈λ≤lim infn→∞Dnn≤lim supn→∞Dnn≤35(7−8log2)≈0.87289, where λ 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(σ)|≤|s−1(s(σ))| for all σ∈Sn, where equality holds if and only if σ=123⋯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 Sn.