A New Identity and Some Applications

Abstract

AbstractLet (n|k) denote the number of k-choices 1≤x1<x2<…<xk≤n satisfying xi-xi-1≥2, i = 2,…, k, n + x1-xk≥2; let (m, n | k) = Σi+j=k (m | i)(n | j). Several elementary proofs of the new identity (m, n|k) = (m + n | k) if 0≤k<m≤n. andif 0≤m≤n, m≤k, are given. Generalizations and applications are considered.