Enumeration of Factor Occurrences in $k$-Bonacci Words over an Infinite Alphabet
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2026
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| _version_ | 1866915908557996032 |
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| author | Ghareghani, Narges Golafshan, Mehdi Mohammad-Noori, Morteza Sharifani, Pouyeh |
| author_facet | Ghareghani, Narges Golafshan, Mehdi Mohammad-Noori, Morteza Sharifani, Pouyeh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01488 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Enumeration of Factor Occurrences in $k$-Bonacci Words over an Infinite Alphabet Ghareghani, Narges Golafshan, Mehdi Mohammad-Noori, Morteza Sharifani, Pouyeh Combinatorics 68R15, 05A15, 11B39, 37B10 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. |
| title | Enumeration of Factor Occurrences in $k$-Bonacci Words over an Infinite Alphabet |
| topic | Combinatorics 68R15, 05A15, 11B39, 37B10 |
| url | https://arxiv.org/abs/2604.01488 |