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

Abstract

This article studies the number of ways of selecting k objects arranged in p circles of sizes n0,...,np−1 such that no two selected ones have less than s objects between them. If ni ≥ sk + 1 for all 0 ≤ i ≤ p − 1, this number is shown to be n0+...+np−2 k n0+...+np−2−sk−1 k−1 . A combinatorial proof of this claim is provided, and two convolution formulas due to Rothe are obtained as corollaries.