A sum rule for $r$-derangements obtained from the Cauchy product of exponential generating functions

TL;DR

This paper introduces a new sum rule for r-derangements, derived from the Cauchy product of exponential generating functions, extending known derangement identities to more restricted cycle structures.

Contribution

It presents a novel sum rule involving binomial coefficients for r-derangements, generalizing classical derangement identities using exponential generating functions.

Findings

Derived a sum rule for r-derangements involving binomial coefficients

Generalized a known relation for derangements to r-derangements

Used Cauchy product of exponential generating functions to prove the identity

Abstract

We propose a sum rule for $r$-derangements (meaning that the elements are restricted to be in distinct cycles in the cycle decomposition) involving binomial coefficients. The identity, obtained using the Cauchy product of two exponential generating functions, generalizes a known relation for usual derangements.