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arXiv 2501.07573math.CO
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矩形和阶梯形表上上升与下降分布的对称性

Symmetry of ascent and descent distributions on rectangular and staircase tableaux

Sergi Elizalde

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AI总结：

本文利用标准 Young 表的箭头编码构造双射，证明矩形与截断阶梯形表上上升、下降分布对称，并推广到统计量保持对合、Sulanke 猜想、rowmotion 及 canon 排列。

摘要：

我们给出了直接的双射证明，证明了在形状为 $λ$ 的标准 Young 表上，上升次数与下降次数的分布具有对称性，其中 $λ$ 是矩形 $(n,n,\dots,n)$ 或截断阶梯形 $(n,n-1,\dots,n-k+1)$。这些情形可视为分级偏序集线性扩张上下降分布更一般对称性的实例；此前 Stanley 和 Farley 对该一般对称性的证明分别基于 $P$-分拆理论和对合原理。在两行矩形 $(n,n)$ 的情形，我们的双射等价于 Dyck 路径上的 Lalanne–Kreweras 对合，从而以双射方式证明了 Narayana 数的对称性。我们的双射通过标准 Young 表的某些箭头编码来定义。这一设置使我们能够在矩形表上构造其他保持统计量的对合，从而为上升与下降在相差一个平移的意义上等分布这一事实提供简单证明，并在三行情形证明了 Sulanke 关于某些统计量的一个猜想。最后，我们利用这些双射在矩形标准 Young 表上定义了一种可能的 rowmotion 概念，并对 canon 排列上下降次数的对称性给出双射证明；canon 排列近期作为 Stirling 排列和拟 Stirling 排列的一种变体得到研究。

英文摘要：

We give direct bijective proofs of the symmetry of the distributions of the number of ascents and descents over standard Young tableaux of shape $λ$, where $λ$ is a rectangle $(n,n,\dots,n)$ or a truncated staircase $(n,n-1,\dots,n-k+1)$. These can be viewed as instances of the more general symmetry of the distribution of descents over linear extensions of graded posets, for which previous proofs by Stanley and Farley were based on the theory of $P$-partitions and the involution principle, respectively. In the case of two-row rectangles $(n,n)$, our bijection is equivalent to the Lalanne--Kreweras involution on Dyck paths, which bijectively proves the symmetry of the Narayana numbers. Our bijections are defined in terms of certain arrow encodings of standard Young tableaux. This setup allows us to construct other statistic-preserving involutions on tableaux of rectangular shape, providing a simple proof of the fact that ascents and descents are equidistributed up to a shift, and proving a conjecture of Sulanke about certain statistics in the case of three rows. Finally, we use our bijections to define a possible notion of rowmotion on standard Young tableaux of rectangular shape, and to give a bijective proof of the symmetry of the number of descents on canon permutations, which have been recently studied as a variation of Stirling and quasi-Stirling permutations.

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