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

2026/07/22 by Vuong Bui
#math.CO

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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≤ B_m-1B_n+1 for every 1≤ m≤ n. Our injection Π_m×Π_n→ Π_m-1×Π_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.

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