Computing the k-binomial complexity of generalized Thue--Morse words

TL;DR

This paper determines the exact k-binomial complexity of generalized Thue--Morse words, confirming a conjecture that it is eventually periodic with a specific period, and introduces new combinatorial tools for analysis.

Contribution

It provides explicit formulas for the k-binomial complexity of generalized TM words and confirms the periodicity conjecture for all k and alphabet sizes.

Findings

Derived explicit formulas for k-binomial complexity functions.

Confirmed the periodicity conjecture with period m^k.

Developed abelian Rauzy graphs as analytical tools.

Abstract

Two finite words are k-binomially equivalent if each subword (i.e., subsequence) of length at most k occurs the same number of times in both words. The k-binomial complexity of an infinite word is a function that maps the integer $n\geq 0$ to the number of k-binomial equivalence classes represented by its factors of length n. The Thue--Morse (TM) word and its generalization to larger alphabets are ubiquitous in mathematics due to their rich combinatorial properties. This work addresses the k-binomial complexities of generalized TM words. Prior research by Lejeune, Leroy, and Rigo determined the k-binomial complexities of the 2-letter TM word. For larger alphabets, work by L\"u, Chen, Wen, and Wu determined the 2-binomial complexity for m-letter TM words, for arbitrary m, but the exact behavior for $k\geq 3$ remained unresolved. They conjectured that the k-binomial complexity function of the m-letter TM word is eventually periodic with period $m^k$. We resolve the conjecture positively by deriving explicit formulae for the k-binomial complexity functions for any generalized TM word. We do this by characterizing k-binomial equivalence among factors of generalized TM words. This comprehensive analysis not only solves the open conjecture, but also develops tools such as abelian Rauzy graphs.