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

TL;DR

This paper derives a formula for counting the number of ways to select k objects from multiple circles with specific spacing constraints, providing combinatorial proofs and formulas.

Contribution

It presents a new explicit formula for counting s-separated k-sets across multiple circles, with a combinatorial proof and additional formulas.

Findings

Derived a closed-form formula for s-separated k-sets in multiple circles.

Provided a combinatorial proof of the main formula.

Developed additional related combinatorial formulas.

Abstract

This article studies the number of ways of selecting $k$ objects arranged in $p$ circles of sizes $n_1,\ldots,n_p$ such that no two selected ones have less than $s$ objects between them. If $n_i\geq sk+1$ for all $1\leq i \leq p$, this number is shown to be $\frac{n_1+\ldots+n_p}{k}\binom{n_1+\ldots+n_p-sk-1}{k-1}$. A combinatorial proof of this claim is provided, and some nice combinatorial formulas are derived.