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

Abstract

We provide a direct injection for the well-known strong log-convexity of the Bell numbers $B_n$, that is $B_m{B}_n\le B_{m-1}{B}_{n+1}$ 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.

6 pages; comments are welcome