On the Number of Restricted One-to-One and Onto Functons Having Integral Coordinates

Let $N_m$ be the set of positive integers $1, 2, \cdots, m$ and $S \subseteq N_m$. In 2000, J. Caumeran and R. Corcino made a thorough investigation on counting restricted functions $f_{|S}$ under each of the following conditions:\begin{itemize}\item[(\textit{a})]$f(a) \leq a$, $\forall a \in S$;\item[(\textit{b})] $f(a) \leq g(a)$, $\forall a \in S$ where $g$ is any nonnegative real-valued continuous functions;\item[(\textit{c})] $g_1(a) \leq f(a) \leq g_2(a)$, $\forall a \in S$, where $g_1$ and $g_2$ are any nonnegative real-valued continuous functions.\end{itemize}Several formulae and identities were also obtained by Caumeran using basic concepts in combinatorics.In this paper, we count those restricted functions under condition $f(a) \leq a$, $\forall a \in S$, which is one-to-one and onto, and establish some formulas and identities parallel to those obtained by J. Caumeran and R. Corcino.