Some Combinatorial Identities

Abstract

For integers n ⩾ 1, k ⩾ 0, and w ⩾ 0, let (n|k) _w denote the number of k-choices

$$ 1\; \le {\rm{ }}{x_1}\; < \;{x_2}\; < \; \cdot \cdot \cdot \; < {x_k}\; \le \;n $$

(1)

from {1,2,3,...,n} satisfying the restrictions

$$ {x_i}\; - \;{x_{i - 1}}\; - \;{\rm{1 }} \ge {\rm{ }}w for i = 2,3, \ldots , k $$

(2)

and

$$ n\;{\rm{ + }}{x_1}\; - \;{x_k}\; - \;{\rm{1 }} \ge {\rm{ }}w. $$