The Ratio Monotonicity of the $q$-Derangement Numbers

TL;DR

This paper proves that the $q$-derangement numbers exhibit a ratio monotone property, a strong form of log-concavity, enhancing understanding of their combinatorial structure.

Contribution

It establishes the ratio monotonicity of $q$-derangement numbers, a property stronger than previously known unimodality and spiral properties.

Findings

$q$-derangement numbers satisfy ratio monotonicity

The property is stronger than log-concavity and unimodality

Enhances understanding of combinatorial number properties

Abstract

We show that the $q$-derangement numbers satisfy a ratio monotone property, which is analogous to the log-concavity and is stronger than the spiral property and the unimodality.