Combinatorial proof of the log-convexity for the derangements in the Coxeter groups

Abstract

We provide the combinatorial proofs of the log-convexity for the derangement numbers in the symmetric group $\mathfrak{S}_n$, hyperoctahedral group $\mathfrak{B}_n$, and the demihyperoctahedral group $\mathfrak{D}_n$. We also show that the sequences of the even and odd derangement numbers in $\mathfrak{S}_n$ and $\mathfrak{B}_n$ are log-convex.