{
  "id": "2412.18425",
  "version": "v1",
  "published": "2024-12-24T13:39:58.000Z",
  "updated": "2024-12-24T13:39:58.000Z",
  "title": "Computing the k-binomial complexity of generalized Thue--Morse words",
  "authors": [
    "M. Golafshan",
    "M. Rigo",
    "M. Whiteland"
  ],
  "comment": "40 pages, 9 figures, submitted for publication",
  "categories": [
    "math.CO",
    "cs.FL"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-24T13:39:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "68R15"
    ],
    "keywords": [
      "generalized thue-morse words",
      "generalized tm word",
      "k-binomial complexity function",
      "m-letter tm word",
      "larger alphabets"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}