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

TL;DR

This paper analyzes the occurrence patterns of digits and factors in the infinite $k$-Bonacci word, providing explicit formulas and recurrence relations for their counts across finite iterations.

Contribution

It introduces closed-form generating functions for digit and factor occurrences in $k$-Bonacci words, revealing Fibonacci-type recurrence structures.

Findings

Closed-form generating functions for digit occurrences

Complete characterization of length-2 factors

Recurrence relations akin to $(k-1)$-step Fibonacci sequences

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.