The number of s-separated k-sets in various circles

Emiliano J. J. Estrugo; Adrián Pastine

paper

The number of s-separated k-sets in various circles

arXiv:1805.01562

Abstract

This article studies the number of ways of selecting $k$ objects arranged in $p$ circles of sizes $n_{1}, \dots, n_{p}$ such that no two selected ones have less than $s$ objects between them. If $n_{i} \geq s k + 1$ for all $1 \leq i \leq p$ , this number is shown to be $\frac{n _{1} + \dots + n _{p}}{k} (k - 1 n _{1} + \dots + n _{p} - s k - 1)$ . A combinatorial proof of this claim is provided, and some nice combinatorial formulas are derived.

7 pages