AI 中文总结
研究为贝尔数强对数凸性提供直接注入，其注入保持划分对中块总数，也是Touchard多项式强\(q\)对数凸性的注入，还应用此注入恢复了相关结果。
AI 中文摘要
我们为贝尔数\(B_n\)著名的强对数凸性提供了一种直接注入，即对于\(1\leq m\leq n\)，有\(B_mB_n\leq B_{m - 1}B_{n + 1}\)。我们的注入\(\Pi_m\times\Pi_n\to \Pi_{m - 1}\times\Pi_{n + 1}\)（其中\(\Pi_n\)表示\([n]\)的所有划分的集合）保持划分对中块的总数。这也是Touchard多项式强\(q\)对数凸性的注入，该结果由Chen、Wang和Yang用解析方法得到。作为注入的应用，我们还恢复了Chern、Diaconis、Kane和Rhoades 的一个相关结果。
英文摘要
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 $Π_m\timesΠ_n\to Π_{m-1}\timesΠ_{n+1}$, where $Π_n$ 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.
Comments6 pages; comments are welcome
