标记二叉树、Catalan排列的子排列和Schur正性
Labeled binary trees, subarrangements of the Catalan arrangements, and Schur positivity
摘要：
1995年，第一作者引入了一个多变量生成函数{ $G$ }，它跟踪标记二叉树中上升和下降的分布。除了证明$G$是对称的，他还证明了$G$是舒尔正的。我们证明了这一猜想通过扩展$G$积极的带状Schur函数。我们使用保权双射来获得这种扩展，其逆的灵感来自于Préville-Ratelle和Viennot的Push-Glide算法。事实上，这种重量保持双射使我们能够建立一个更强的版本的第一作者的猜想表明，生成函数限制到标记的二叉树与固定的树冠仍然是舒尔积极的。我们还讨论了超平面安排的设置中的应用。我们表明，一定的专业化$G$等于Frobenius特性的自然$\mathfrak{S}_n$ -行动的半序安排，我们然后扩展的{Frobenius特性}的福克斯字符。我们还构造了一个$\mathfrak{S}_n$ -行动的区域的线性安排使用一组树研究Bernardi，并随后计算的字符，这一行动采用拉格朗日反演。由此产生的表达式概括了Postnikov的公式中的Linial安排的区域的数量。作为最后的应用，我们证明了$γ$ -非负的分布的右边缘的数量在本地二叉搜索树。
英文摘要：
In 1995, the first author introduced a multivariate generating function {$G$} that tracks the distribution of ascents and descents in labeled binary trees. In addition to proving that $G$ is symmetric, he conjectured that $G$ is Schur positive. We prove this conjecture by expanding $G$ positively in terms of ribbon Schur functions. We obtain this expansion using a weight-preserving bijection whose inverse is inspired by the Push-Glide algorithm of Préville-Ratelle and Viennot. In fact, this weight-preserving bijection allows us to establish a stronger version of the first author's conjecture showing that the generating function restricted to labeled binary trees with a fixed canopy is still Schur positive. We also discuss applications in the setting of hyperplane arrangements. We show that a certain specialization of $G$ equals the Frobenius characteristic of the natural $\mathfrak{S}_n$-action on regions of the semiorder arrangement, which we then expand in terms of {the Frobenius characteristics} of Foulkes characters. We also construct an $\mathfrak{S}_n$-action on regions of the Linial arrangement using a set of trees studied by Bernardi, and subsequently compute the character of this action by employing Lagrange inversion. The resulting expression generalizes Postnikov's formula for the number of regions in the Linial arrangement. As a final application, we prove $γ$-nonnegativity for the distribution of the number of right edges over local binary search trees.
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