Connections between certain numbers related to derangements and $r$-permutations

Abstract

For non-negative integer parameters $r,u,m,n$ define \begin{align*} \cal{D}(r,u,m,n) := \big\{\ \sigma\in \cal{S}_{r+n}\ \big|\ \sigma(x)=y \textrm{ for exactly } u \textrm{ pairs } (x,y) \textrm{ such that } 1\leq x,y\leq r \textrm{ and } \sigma(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\{\ \sigma\in \cal{S}_{r+n}\ \big|\ \forall_{1\leq x<y\leq r} \ x \textrm{ and } y \textrm{ are in disjoint cycles of } \sigma \textrm{ and } \sigma(z)=z \textrm{ for exactly } u \textrm{ elements } 1\leq z\leq r, \textrm{ and } \sigma(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.