A problem of enumeration of two-color bracelets with several variations

TL;DR

This paper explores the enumeration of two-color bracelets with fixed black beads, introduces variations of the problem, and provides recursion formulas for counting bracelets with three or more colors.

Contribution

It presents new enumeration formulas for two-color bracelets with variations and extends the approach to t-color bracelets with recursion formulas.

Findings

Derived formulas for counting two-color bracelets with fixed black beads.

Introduced natural variations of the bracelet enumeration problem.

Provided recursion formulas for counting t-color bracelets for t ≥ 3.

Abstract

We consider the problem of enumeration of incongruent two-color bracelets of $n$ beads, $k$ of which are black, and study several natural variations of this problem. We also give recursion formulas for enumeration of $t$-color bracelets, $t\geq3.