A Pascal-like Bound for the Number of Necklaces with Fixed Density

TL;DR

This paper introduces a Pascal-like bound for counting binary necklaces with fixed density, extending to k-ary necklaces, with applications in Nichols algebras of diagonal type.

Contribution

It presents a novel combinatorial bound for necklaces with fixed density, generalizing to k-ary cases and connecting to algebraic structures.

Findings

Derived a Pascal-like bound for binary necklaces with fixed density

Extended the bound to k-ary necklaces and Lyndon words with fixed content

Applied the bound to Nichols algebras of diagonal type

Abstract

A bound resembling Pascal's identity is presented for binary necklaces with fixed density using Lyndon words with fixed density. The result is generalized to k-ary necklaces and Lyndon words with fixed content. The bound arises in the study of Nichols algebras of diagonal type.