On the number of combinations without certain separations

TL;DR

This paper derives formulas for counting the number of ways to select objects from a line or circle with restrictions on the separation between chosen objects, extending combinatorial enumeration methods.

Contribution

It introduces three formulas for line arrangements and a closed-form expression for circular arrangements under specific separation constraints.

Findings

Derived three formulas for line arrangements with separation restrictions.

Established a closed-form formula for circular arrangements with separation constraints.

Extended combinatorial enumeration techniques for constrained selections.

Abstract

In this paper we enumerate the number of ways of selecting $k$ objects from $n$ objects arrayed in a line such that no two selected ones are separated by $m-1,2m-1,...,pm-1$ objects and provide three different formulas when $m,p\geq 1$ and $n\geq pm(k-1)$. Also, we prove that the number of ways of selecting $k$ objects from $n$ objects arrayed in a circle such that no two selected ones are separated by $m-1,2m-1,...,pm-1$ objects is given by $\frac{n}{n-pk}\binom{n-pk}{k}$, where $m,p\geq 1$ and $n\geq mpk+1$.