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arXiv · 1804.00992

Different classes of binary necklaces and a combinatorial method for their enumerations

Romeo Meštrović
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Abstract

In this paper we investigate enumeration of some classes of $n$-character strings and binary necklaces. Recall that binary necklaces are necklaces in two colors with length $n$. We prove three results (Theorems 1, 1' and 2) concerning the numbers of three classes of $j$-character strings (closely related to some classes of binary necklaces or Lyndon words). Using these results, we deduce Moreau's necklace-counting function for binary aperiodic necklaces of length $k$ \cite{mo} (Theorem 3), and we prove the binary case of MacMahon's formula from 1892 \cite{ma} (also called Witt's formula) for the number of necklaces (Theorem 4). Notice that we give proofs of Theorems 3 and 4 without use of Burnside's lemma and Pólya enumeration theorem. Namely, the methods used in our proofs of auxiliary and main results presented in Sections 3 and 4 are combinatorial in spirit and they are based on counting method and some facts from elementary number theory.

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Romeo Meštrović. 2018-03-31. Different classes of binary necklaces and a combinatorial method for their enumerations. https://arxiv.org/abs/1804.00992

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