Analytic properties arising from the Baxter numbers

TL;DR

This work investigates analytic properties of Baxter numbers by embedding Baxter-related polynomials into the $(1,t)$-Hoggatt sums and proving interlacing via Hadamard-product techniques, establishing that Baxter polynomials $PB_n(t)$ form a Sturm sequence. It further analyzes asymptotic $r$-log-convexity of the Baxter numbers $B_n$ using $P$-recursive asymptotics, showing asymptotic 1-log-convexity and providing symbolic evidence for higher-order log-convexity, including $2$- and $3$-log-convexity thresholds. The paper also proves that, for fixed $m\ge2$, the Hoggatt polynomials $H_n^{[m]}(1,t)$ form Sturm sequences, extending real-rootedness results and yielding interlacing relations among successive polynomials. Three conjectures connect these results to broader families of $(q,t)$-Hoggatt sums and the infinite log-convexity landscape of Baxter-related sequences, highlighting deep links between combinatorics, real-rooted polynomial theory, and asymptotic analysis.

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.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: Garloff-Wagner-1996
• Corollary 2.1
• proof
• Theorem 2.2: Markov-1892
• proof : Proof of Theorem \ref{['thm:interlace-HP']}
• Theorem 3.1: Hou-Zhang2019
• proof : Proof of Theorem \ref{['thm:asym-r-lgcvBn']}