Enumeration of derangements with descents in prescribed positions

TL;DR

This paper provides a combinatorial proof and explicit formulas for counting derangements with descents at specific positions, extending previous generating function results and exploring correlations between permutation properties.

Contribution

It offers a new combinatorial proof, explicit enumeration formulas, and a generalization of Euler's difference tables for derangements with prescribed descents.

Findings

Combinatorial proof of Han and Xin's generating function

Explicit formulas for derangements with descents in fixed positions

Positive correlation between derangement and descent events in permutations

Abstract

We enumerate derangements with descents in prescribed positions. A generating function was given by Guo-Niu Han and Guoce Xin in 2007. We give a combinatorial proof of this result, and derive several explicit formulas. To this end, we consider fixed point $\lambda$-coloured permutations, which are easily enumerated. Several formulae regarding these numbers are given, as well as a generalisation of Euler's difference tables. We also prove that except in a trivial special case, if a permutation $\pi$ is chosen uniformly among all permutations on $n$ elements, the events that $\pi$ has descents in a set $S$ of positions, and that $\pi$ is a derangement, are positively correlated.