Troupes, cumulants, and stack-sorting

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