Counting 3-stack-sortable permutations

Abstract

We prove a "decomposition lemma" that allows us to count preimages of certain sets of permutations under West's stack-sorting map s. As a first application, we give a new proof of Zeilberger's formula for the number W2(n) of 2-stack-sortable permutations in Sn. Our proof generalizes, allowing us to find an algebraic equation satisfied by the generating function that counts 2-stack-sortable permutations according to length, number of descents, and number of peaks. This is also the first proof of this formula that generalizes to the setting of 3-stack-sortable permutations. Indeed, the same method allows us to obtain a recurrence relation for W3(n), the number of 3-stacksortable permutations in Sn. Hence, we obtain the first polynomial-time algorithm for computing these numbers. We compute W3(n) for n ≤ 174, vastly extending the 13 terms of this sequence that were known before. We also prove the first nontrivial lower bound for lim n→∞ W3(n) 1/n , showing that it is at least 8.659702. Invoking a result of Kremer, we also prove that lim n→∞ Wt(n) 1/n ≥ ( √ t + 1) 2 for all t ≥ 1, which we use to improve a result of Smith concerning a variant of the stack-sorting procedure. Our computations allow us to disprove a conjecture of Bóna, although we do not yet know for sure which one.In fact, we can refine our methods to obtain a recurrence for W3(n, k, p), the number of 3-stacksortable permutations in Sn with k descents and p peaks. This allows us to gain a large amount of evidence supporting a real-rootedness conjecture of Bóna. Using part of the theory of valid hook configurations, we give a new proof of a γ-nonnegativity result of Brändén, which in turn implies an older result of Bóna. We then answer a question of the current author by producing a set A ⊆ S11 such that σ∈s −1 (A) x des(σ) has nonreal roots. We interpret this as partial evidence against the same real-rootedness conjecture of Bóna that we found evidence supporting. Examining the parities of the numbers W3(n), we obtain strong evidence against yet another conjecture of Bóna. We end with some conjectures of our own.

Quantifying Noninvertibility in Discrete Dynamical Systems

“…If π is a permutation with largest entry m, then we can write π = LmR for some permutations L and R. Then For each positive integer n, we can view s as a discrete dynamical system on S n . We refer the reader to [2,3,6,8] and the references therein for more information about this map. In [6][7][8], the first author found methods for computing the number of preimages of an arbitrary permutation under the stack-sorting map.…”

“…We refer the reader to [2,3,6,8] and the references therein for more information about this map. In [6][7][8], the first author found methods for computing the number of preimages of an arbitrary permutation under the stack-sorting map. Unfortunately, it seems quite difficult to use these methods in order to find an explicit formula for deg(s : S n → S n ).…”

Catalan intervals and uniquely sorted permutations

“…This map, denoted by s, sends permutations of length n to permutations of length n. It is a slight variant of the stack-sorting algorithm that Knuth introduced in [27]. The map s was studied extensively in West's 1990 Ph.D. thesis [44] and has received a considerable amount of attention ever since [5,6,8,14]. We give necessary background results concerning the stack-sorting map in Section 3, but the reader seeking additional historical motivation should consult [5,6,14] and the references therein.…”

Stack words and a bound for 3-stack sortable permutations

“…to prove, but now has several complicated proofs. See [10] , [7], [6] or [5] for various proofs. Theorem 1.2.…”

“…For t = 3, he proved the upper bound lim n→∞ n W 3 (n) ≤ 12.53296, and for t = 4, he proved the upper bound lim n→∞ n W 3 (n) ≤ 21.97225. Results on related lower bounds can be found in Defant's paper [5], where it is shown that lim n→∞ n W 3 (n) ≥ 8.659702 and that lim n→∞ n W t (n) ≥ ( √ t + 1) 2 , along with a new proof for the formula for W 2 (n), and a polynomial time algorithm to compute the numbers W 3 (n).…”