有界重复单词上的栈排序
Stack Sorting on Words with Bounded-Repetition
AI总结：
本文研究有界重复栈排序算子族，给出二元单词上的显式公式，证明分离-加速现象，并揭示算子族不介于单次与无限次重复之间。
AI中文摘要：
我们考虑栈排序算子族 (s_m)_{m >= 1} 作用于单词的规则如下：s_m(w) 是对输入单词 w 应用通常的 West 栈排序算法所产生的输出，但允许同一字母在栈中连续堆叠至多 m 次。我们的第一个主要定理给出了算子 s_m 在二元单词 w = a^p b a^q（其中不同字母 a > b）上的作用的显式公式：s_m(a^p b a^q) = a^max(p-m,0) b a^min(p,m)+q，以及 d_m(a^k b) = ceil(k/m)。由上述公式可知，对于每个 m < m'，在二元单词 w = a^k b 上，我们有 d_m(w) >= d_{m'}(w)，而且比值 d_m(w)/d_{m'}(w) 总是可以恰好取到 m'/m，因此在该指定类上，比值 d_m/d_{m'} 是无界的——这是预期的分离-加速现象的显式形式。此外，我们提供了长度为 4 的单词的显式构造，使得 s_m 和 s_{m'} 不可交换，以及长度为 7 的单词，使得速度 d_m 在 m 上不单调。进一步，我们给出了一个完整的递归刻画——推广了 Knuth 和 West 的经典 231-避免结果——刻画了仅使用 s_m 一次即可排序的单词，然后利用它表明，对于单遍可排序单词，不存在与 m 无关的经典模式避免条件。最后，我们给出了不等式 d_1(w) >= d_m(w) >= d_infinity(w) 对所有 m 在双字母单词上成立的完整证明，以及一个结构结果，在一般情况下对单遍建立该不等式；然而，我们表明算子族 (s_m) 并不完全位于 s_1 和 s_infinity 之间，对每个 n >= 7 展示了显式单词 w_n，使得 d_1(w_n) = n-4 < n-3 = d_m(w_n) 对所有 m >= 2（包括 m = infinity）成立，因此上述不等式在一般情况下不成立。
英文摘要：
We consider the family of operators of stack sorting (s_m)_{m >= 1} acting on words according to the following rule: s_m(w) is the output produced by the usual West algorithm of stack sorting on the input word w, but allowing for at most m repetitions of the same letter stacked in succession. Our first main theorem gives an explicit formula for the action of the operator s_m on binary words w = a^p b a^q (with distinct letters a > b): s_m(a^p b a^q) = a^max(p-m,0) b a^min(p,m)+q, d_m(a^k b) = ceil(k/m). From the above formulas it follows that for every m < m' we have d_m(w) >= d_{m'}(w) for binary words w = a^k b, and moreover, the ratio d_m(w)/d_{m'}(w) is always possible to be chosen exactly m'/m, so the ratios d_m/d_{m'} are unbounded for this family on the specified class -- this is an explicit form of the expected separation-speedup phenomenon. Additionally, we provide explicit constructions of words of length 4 for which s_m and s_{m'} do not commute, and words of length 7 for which the speed d_m is not monotonic in m. Further, we provide a full recursive characterization -- generalizing the classical 231-avoidance result of Knuth and West -- of words which can be sorted using s_m only once, and then we employ it to show that there cannot exist an m-independent classical pattern avoidance condition for one-pass sortable words. Finally, we give a complete proof of the inequality d_1(w) >= d_m(w) >= d_infinity(w) for all m for two-letter words, together with a structural result establishing it for a single pass in general; nevertheless, we show that the family of operators (s_m) does not, after all, lie entirely between s_1 and s_infinity, exhibiting for every n >= 7 an explicit word w_n with d_1(w_n) = n-4 < n-3 = d_m(w_n) for every m >= 2, including m = infinity, so that the inequality above is false in general.
