Troupes, cumulants, and stack-sorting

Abstract

In several cases, a sequence of free cumulants that counts certain binary plane trees corresponds to a sequence of classical cumulants that counts the decreasing versions of the same trees. Using two new operations on binary plane trees that we call insertion and decomposition, we prove that this surprising phenomenon holds for families of trees that we call troupes. We give a simple characterization of troupes, showing that they are plentiful. Troupes provide a broad framework for generalizing several of the results that are known about West's stack-sorting map s. Indeed, we give new proofs of some of the main theorems underlying techniques that have been developed recently for understanding s; these new proofs are far more conceptual than the original ones, explain how the objects called valid hook configurations arise very naturally, and generalize to the context of troupes. To illustrate these general techniques, we enumerate 2-stack-sortable and 3-stack-sortable alternating permutations of odd length and 2-stack-sortable and 3-stack-sortable permutations whose descents are all peaks.The unexpected connection between troupes and cumulants provides a powerful new tool for analyzing the stack-sorting map that hinges on free probability theory. We give numerous applications of this method. For example, we show that if σ∈Sn−1 is chosen uniformly at random and des denotes the descent statistic, then the expected value of des(s(σ))+1 is(3−∑j=0n1j!)n. Furthermore, the variance of des(s(σ))+1 is asymptotically (2+2e−e2)n. We obtain similar results concerning the expected number of descents of postorder readings of decreasing binary plane trees of various types. We also obtain improved estimates for |s(Sn)| and an improved lower bound for the degree of noninvertibility of s:Sn→Sn. The combinatorics of valid hook configurations allows us to give two novel formulas that convert from free to classical (univariate) cumulants. The first formula is given by a sum over noncrossing partitions, and the second is given by a sum over 231-avoiding valid hook configurations. We pose several conjectures and open problems.