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 $W_2(n)$ of 2-stack-sortable permutations in $S_n$. 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 $W_3(n)$, the number of 3-stack-sortable permutations in $S_n$. Hence, we obtain the first polynomial-time algorithm for computing these numbers. We compute $W_3(n)$ for $n\le 174$, vastly extending the 13 terms of this sequence that were known before. We also prove the first nontrivial lower bound for $\underset{n\to \infty }{\lim }{W}_3{(n)}^{1/n}$, showing that it is at least 8.659702. Invoking a result of Kremer, we also prove that $\underset{n\to \infty }{\lim }{W}_t{(n)}^{1/n}\ge {(\sqrt{t}+1)}^2$ for all $t\ge 1$, 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 $W_3(n,k,p)$, the number of 3-stack-sortable permutations in $S_n$ 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 $A\subseteq S_{11}$ such that ${\sum }_{\sigma \in s^{-1}(A)}{x}^{\mathrm{des}(\sigma )}$ 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 $W_3(n)$, we obtain strong evidence against yet another conjecture of Bóna. We end with some conjectures of our own.

我们证明了一个“分解引理”，它使我们能够计算West的堆栈排序图s下某些置换集合的原像。作为第一个应用，我们给出Zeilberger公式的新证明${\mathrm{w\; ^}}_2（ñ）$ 中的2堆栈可排序排列 ${\mathrm{小号}}_{ñ}$。我们的证明是广义的，使我们能够找到生成函数满足的代数方程式，该函数根据长度，下降次数和峰数对2个堆栈可排序的排列进行计数。这也是该公式推广到3堆可排序排列设置的第一个证明。确实，相同的方法允许我们获得以下项的递归关系：${\mathrm{w\; ^}}_3（ñ）$，其中的3个堆栈可排序排列的数量 ${\mathrm{小号}}_{ñ}$。因此，我们获得了用于计算这些数字的第一个多项式时间算法。我们计算${\mathrm{w\; ^}}_3（ñ）$ 对于 $ñ\le 174$，大大扩展了此序列中以前已知的13个术语。我们还证明了第一个非平凡的下界$\underset{ñ\to \infty }{\mathrm{林}}{\mathrm{w\; ^}}_3{（ñ）}^{\mathrm{1个}/ñ}$，表明它至少是8.659702。调用Kremer的结果，我们还证明$\underset{ñ\to \infty }{\mathrm{林}}{\mathrm{w\; ^}}_{Ť}{（ñ）}^{\mathrm{1个}/ñ}\ge {（\sqrt{Ť}+\mathrm{1个}）}^2$ 对全部 $Ť\ge \mathrm{1个}$，我们用它来改善Smith提出的有关堆栈排序过程变体的结果。我们的计算使我们可以反驳伯纳的一个猜想，尽管我们尚不确定哪一个。

实际上，我们可以完善我们的方法来获得 ${\mathrm{w\; ^}}_3（ñ，ķ，p）$，其中的3个堆栈可排序排列的数量 ${\mathrm{小号}}_{ñ}$有k个下降峰和p个峰。这使我们获得了大量证据来支持博纳的真实根源推测。使用有效钩子配置理论的一部分，我们给出了Brändén的γ负性结果的新证明，而这又暗示了Bóna的较早结果。然后，我们通过制作一组答案来回答当前作者的问题$\mathrm{一种}\subseteq {\mathrm{小号}}_{11}$ 这样 ${\sum }_{\sigma \in s^{-\mathrm{1个}}（\mathrm{一种}）}{X}^{\mathrm{des}（\sigma ）}$有非真实的根源。我们将其解释为部分证据，以反对我们发现证据支持的波那真实根源性推测。检查数字的奇偶性${\mathrm{w\; ^}}_3（ñ）$，我们获得了强有力的证据来反对波纳的另一种猜想。我们以自己的一些猜想结束。