Derangements and Relative Derangements of Type $B$

TL;DR

This paper introduces signed relative derangements of type B, providing a combinatorial interpretation and a type B analogue of the relation between relative derangements and classical derangements.

Contribution

It defines signed relative derangements of type B and offers a combinatorial interpretation using signed skew derangements, extending classical derangement relations.

Findings

Established a type B analogue of the relation between relative and classical derangements.

Provided a combinatorial interpretation using signed skew derangements.

Connected the concept to signed permutations and derangement theory.

Abstract

By introducing the notion of relative derangements of type $B$, also called signed relative derangements, which are defined in terms of signed permutations, we obtain a type $B$ analogue of the well-known relation between relative derangements and the classical derangements. While this fact can be proved by using the principle of inclusion and exclusion, we present a combinatorial interpretation with the aid of the intermediate structure of signed skew derangements.