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arXiv · 2607.19803

A direct injection for the strong $q$-log-convexity of Touchard polynomials

Vuong Bui
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Abstract

We provide a direct injection for the well-known strong log-convexity of the Bell numbers $B_n$, that is $B_mB_n\le B_{m-1}B_{n+1}$ for every $1\le m\le n$. Our injection $\Pi_m\times\Pi_n\to \Pi_{m-1}\times\Pi_{n+1}$, where $\Pi_n$ denotes the set of all partitions of $[n]$, preserves the total number of blocks in the pair of partitions. In other words, it is also an injection for the strong $q$-log-convexity of Touchard polynomials, a result established by Chen, Wang, and Yang using analytical arguments. As an application of the injection, we also recover a related result of Chern, Diaconis, Kane, and Rhoades.

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Vuong Bui. 2026-07-22. A direct injection for the strong $q$-log-convexity of Touchard polynomials. https://arxiv.org/abs/2607.19803

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