Labeled binary trees, subarrangements of the Catalan arrangements, and Schur positivity

Abstract

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 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 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.

The Bernardi Formula for Nontransitive Deformations of the Braid Arrangement

“…Example 8. The tree in Figure 2 has cadet sequences (4, 5, 1), (2, 3), and (6). Subsequences of these which are convex with respect to the relation "is the cadet of", for example (4, 5) and (3), are also cadet sequences.…”

“…. , j − 2} for 1 = j ∈[6]. Leaves which lie to the right of 6, above 3, or above 5 are omitted from the left picture for clarity.…”

Euler–Catalan’s Number Triangle and Its Application

“…Let us consider the following combinatorial object: a labeled binary tree of size n with m ascents on the left branch. In such a tree structure, a tree consists of n nodes, each node has at most two children (the left child and the right child), and each node has its own label (a unique value from 1 to n) [14]. By the left branch of a tree, we mean the tree obtained by removing all right children (the identifier St000061 in [15]).…”

Bijections for Faces of the Shi and Catalan Arrangements

“…There are several interesting examples, including the braid, Catalan, and Shi arrangements (defined below). These arrangements have been studied in numerous research papers, such as [2,6,7,22,17,24,19,18,11,8,16,5]. However, one aspect of these arrangements that has received less attention is their faces, which are the focus of this paper.…”