Asymptotic normality arising in Baxter permutations

TL;DR

This paper proves the asymptotic normality of refined Baxter numbers, which count various combinatorial objects, using recurrence relations and symbolic computation, advancing understanding of Baxter permutations' distribution.

Contribution

It establishes the asymptotic normality of refined Baxter numbers through a semi-automatic method involving recurrence relations and symbolic computation.

Findings

Proved asymptotic normality of refined Baxter numbers.

Utilized recurrence relations and symbolic computation.

Addressed computational challenges due to lack of closed-form expressions.

Abstract

Baxter permutations arose in the study of fixed points of the composite of commuting functions by Glen Baxter in 1964. This type of permutations are counted by Baxter numbers $B_n$. It turns out that $B_n$ enumerate a lot of discrete objects such as the bases for subalgebras of the Malvenuto-Reutenauer Hopf algebra, the pairs of twin binary trees on $n$ nodes, or the diagonal rectangulations of an $n\times n$ grid. The refined Baxter number $D_{n,k}$ also count many interesting objects including the Baxter permutations of $n$ with $k-1$ descents and $n-k$ rises, twin pairs of binary trees with $k$ left leaves and $n-k+1$ right leaves, or plane bipolar orientations with $k+1$ faces and $n-k+2$ vertices. In this paper, we obtain the asymptotic normality of the refined Baxter number $D_{n,k}$ by using a sufficient condition due to Bender. In the course of our proof, the computation involving $B_n$ and some related numbers is crucial, while $B_n$ has no closed form which make the computation untractable. To address this problem, we employ the method of asymptotics of the solutions of linear recurrence equations. Our proof is semi-automatic. All the asymptotic expansions and recurrence relations are proved by utilizing symbolic computation packages.