Connections between certain numbers related to derangements and $r$-permutations
Piotr Miska, Błażej Żmija
TL;DR
This work introduces two refined families of derangement-type permutation counts, $\mathcal{D}(r,u,m,n)$ and $\mathcal{D}_{r,u,m}(n)$, and establishes deep connections to $r$-derangements and generalized rencontres numbers through purely combinatorial counting. A cycle-splitting main lemma provides a structural mapping between these sets and enables exact relations such as $d_k(r,u,m,n)=\left[rk\right]d_{r,u,m}(n)$ and $d(r,u,m,n)=r!\,d_{r,u,m}(n)$, with parity-sensitive refinements. The paper also presents a power-series approach yielding closed-form generating functions, and reduces the generalized counts to the classical $r$-derangement numbers via a simple binomial-factor decomposition. Finally, explicit formulas and exponential generating functions for the numbers of even/odd $r$-derangements are derived, including exact expressions and recurrences, broadening the understanding of derangement-like structures and their algebraic encodings.
Abstract
For non-negative integer parameters $r,u,m,n$ define \begin{align*} \cal{D}(r,u,m,n) := \big\{\ σ\in \cal{S}_{r+n}\ \big|\ σ(x)=y \textrm{ for exactly } u \textrm{ pairs } (x,y) \textrm{ such that } 1\leq x,y\leq r \textrm{ and } σ(t)=t \textrm{ for exactly } m \textrm{ elements } r+1\leq t\leq r+n\ \big\} \end{align*} and \begin{align*} \cal{D}_{r,u,m}(n) := \big\{\ σ\in \cal{S}_{r+n}\ \big|\ \forall_{1\leq x<y\leq r} \ x \textrm{ and } y \textrm{ are in disjoint cycles of } σ\textrm{ and } σ(z)=z \textrm{ for exactly } u \textrm{ elements } 1\leq z\leq r, \textrm{ and } σ(t)=t \textrm{ for exactly } m \textrm{ elements } r+1\leq t\leq r+n\ \big\}, \end{align*} where $\mathcal{S}_{n}$ denotes the set of all the permutations of $\{1,\ldots ,n\}$. In this paper we study connections between the sets $\mathcal{D}(r,u,m,n)$, $\mathcal{D}_{r,u,m}(n)$, and the sets of (some classes of) $r$-derangements. We rely mostly on counting arguments.