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Counting 3-stack-sortable permutations
Journal of Combinatorial Theory Series A ( IF 0.9 ) Pub Date : 2020-01-21 , DOI: 10.1016/j.jcta.2020.105209
Colin Defant

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-stack-sortable permutations in Sn. Hence, we obtain the first polynomial-time algorithm for computing these numbers. We compute W3(n) for n174, vastly extending the 13 terms of this sequence that were known before. We also prove the first nontrivial lower bound for limnW3(n)1/n, showing that it is at least 8.659702. Invoking a result of Kremer, we also prove that limnWt(n)1/n(t+1)2 for all t1, 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-stack-sortable 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 AS11 such that σs1(A)xdes(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.



中文翻译:

计算3个堆栈可排序的排列

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w ^2ñ 中的2堆栈可排序排列 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:w ^3ñ,其中的3个堆栈可排序排列的数量 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w ^3ñ 对于 ñ174,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ñw ^3ñ1个/ñ,表明它至少是8.659702。调用Kremer的结果,我们还证明ñw ^Ťñ1个/ñŤ+1个2 对全部 Ť1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 w ^3ñķp,其中的3个堆栈可排序排列的数量 小号ñk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一种小号11 这样 σs-1个一种Xdesσ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w ^3ñ,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。

更新日期:2020-01-21
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