Analytic properties arising from the Baxter numbers

TL;DR

This paper investigates the analytic properties of Baxter numbers, demonstrating interlacing zeros of descent polynomials, proving asymptotic $r$-log-convexity, and confirming $2$-log-convexity through symbolic computation, thus advancing understanding in combinatorics and algebra.

Contribution

It introduces new interlacing zero properties of Baxter permutation polynomials and establishes their asymptotic $r$-log-convexity, using $(q,t)$-Hoggatt sums and symbolic computation.

Findings

Descent polynomials of Baxter permutations have interlacing zeros.

Baxter numbers are asymptotically $r$-log-convex.

Confirmed $2$-log-convexity of Baxter numbers using symbolic computation.

Abstract

Baxter numbers are known as the enumeration of Baxter permutations and numerous other discrete structures, playing a significant role across combinatorics, algebra, and analysis. In this paper, we focus on the analytic properties related to Baxter numbers. We prove that the descent polynomials of Baxter permutations have interlacing zeros, which is a property stronger than real-rootedness. Our approach is based on Dilks' framework of $(q,t)$-Hoggatt sums, which is a $q$-analog for Baxter permutations. Within this framework, we show that the family of $(1,t)$-Hoggatt sums satisfies the interlacing property using fundamental results on Hadamard products of polynomials. For Baxter numbers, we prove their asymptotic $r$-log-convexity via asymptotic expansions of $P$-recursive sequences. In particular, we confirm their $2$-log-convexity using symbolic computation techniques.