(untitled) (European Journal of Pure and Applied Mathematics, 2023)
Let N_m be the set of positive integers 1, 2, …, m and S 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: itemize [(a)]f(a) ≤ a, a S; [(b)] f(a) ≤ g(a), a S where g is any nonnegative real-valued continuous functions; [(c)] g_1(a) ≤ f(a) ≤ g_2(a), a S, where g_1 and g_2 are any nonnegative real-valued continuous functions. 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) ≤ a, a S, which is one-to-one and onto, and establish some formulas and identities parallel to those obtained by J. Caumeran and R. Corcino.