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

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.