Efficient Methods in Counting Generalized Necklaces

TL;DR

This paper explores efficient computational methods for counting generalized necklaces, aiming to reduce the intensive calculations involved in existing formulas by proposing alternative evaluation strategies.

Contribution

It introduces alternative approaches to evaluate the generalized necklace counting formula more efficiently, reducing computational complexity.

Findings

Proposes methods to amortize computational costs

Analyzes the complexity of existing formulas

Suggests more efficient evaluation techniques

Abstract

It is shown in [7] by Venkaiah in 2015 that a category of the number of generalized can be computed using the expression \begin{equation*} e(n, q) = \frac{1}{(q-1) ord(\lambda) n} \sum^{ord(\lambda)n}_{\substack{t \in \mathbb{F}_q \setminus \{0\}, i=1 \\ t^{\frac{n}{\gcd(n, i)}} \lambda^{\frac{i}{\gcd(n,i)}} = 1}}(q^{\gcd(n,i)} - 1) + 1 \end{equation*} where $q$ (number of colors) is the size of the prime field $\mathbb{F}_q$, $\lambda$ is the constant of the consta-cyclic shift, $n$ is the length of the necklace. However, direct evaluation of this expression requires, apart from the $\gcd$ computations, $2*(q-1)*Ord(\lambda)*n$ exponentiations and $(q-1)*Ord(\lambda)*n$ multiplications, at most $(q-1)*Ord(\lambda)*n$ exponentiations and at most $2*(q-1)*Ord(\lambda)*n$ additions and hence computationally intensive. This note discusses various other ways of evaluating the expression and tries to throw some light on amortizing the amount of computation.