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

TL;DR

This paper presents combinatorial proofs demonstrating the log-convexity of derangement numbers across various Coxeter groups, including symmetric, hyperoctahedral, and demihyperoctahedral groups, and explores properties of even and odd derangement sequences.

Contribution

It provides the first combinatorial proofs of log-convexity for derangement numbers in these Coxeter groups, extending known results.

Findings

Log-convexity of derangement numbers in symmetric, hyperoctahedral, and demihyperoctahedral groups

Log-convexity of even and odd derangement sequences in symmetric and hyperoctahedral groups

Combinatorial proofs establishing these properties

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.