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Enumeration of Factor Occurrences in $k$-Bonacci Words over an Infinite Alphabet

Narges Ghareghani, Mehdi Golafshan, Morteza Mohammad-Noori, Pouyeh Sharifani

Abstract

We study the $k$-Bonacci word over the infinite alphabet $\mathbb{N}$. Since the alphabet is infinite, the usual factor complexity is infinite and does not provide any information. We therefore investigate factor occurrence statistics in the finite iterates. For $k \ge 3$, we obtain closed forms for the generating functions (with respect to the iteration index) that count the number of occurrences of an arbitrary digit in the $n$th iterate. We then characterize the complete set of length-$2$ factors occurring in the infinite word and compute, for each such factor, a closed form for the generating function encoding its number of occurrences across all finite iterates. As a consequence, the associated counting sequences satisfy uniform $(k\!-\!1)$-step Fibonacci-type recurrences and admit a description in terms of $(k\!-\!1)$-Bonacci enumeration phenomena, including self-convolution structures.

Enumeration of Factor Occurrences in $k$-Bonacci Words over an Infinite Alphabet

Abstract

We study the -Bonacci word over the infinite alphabet . Since the alphabet is infinite, the usual factor complexity is infinite and does not provide any information. We therefore investigate factor occurrence statistics in the finite iterates. For , we obtain closed forms for the generating functions (with respect to the iteration index) that count the number of occurrences of an arbitrary digit in the th iterate. We then characterize the complete set of length- factors occurring in the infinite word and compute, for each such factor, a closed form for the generating function encoding its number of occurrences across all finite iterates. As a consequence, the associated counting sequences satisfy uniform -step Fibonacci-type recurrences and admit a description in terms of -Bonacci enumeration phenomena, including self-convolution structures.

Paper Structure

This paper contains 15 sections, 19 theorems, 46 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Related work
  4. Notations and conventions
  5. Preliminaries
  6. Factor Occurrence Counts and the Shift Identity
  7. Digit Occurrence Generating Functions
  8. Length-2 Factor Occurrences
  9. Classification into families
  10. Generating functions by family
  11. Concluding Remarks
  12. Numerical Tables for k=4
  13. Digit-occurrence counts
  14. Length-2 factors in B1(4) (sample counts)
  15. Mixed selection of length-2 factors (sample counts)

Key Result

Lemma 1

Let $k \ge 2$ and let $n \in \mathbb{N}_{>0}$. Then $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (40)

  • Lemma 1: M
  • Definition 2: M
  • Remark
  • Lemma 3: M
  • Theorem 4
  • proof
  • Corollary 5
  • proof
  • Lemma 6
  • proof
  • ...and 30 more