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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.20018 |
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| _version_ | 1866908380202795008 |
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| author | Caalim, Jonathan Tanaka, Yu-ichi |
| author_facet | Caalim, Jonathan Tanaka, Yu-ichi |
| contents | A sequence $(e_i)_{i \le m}$ of nonnegative integers $e_i$, where $m \in \mathbb{N}$ or $m =\infty$, is called a binomid index if $\sum_{i=n-k+1}^{n} e_i\geq \sum_{i=1}^ke_i$ for all $k, n \in \mathbb{N}$ such that $ 1\le k \le n < m$. Infinite binomid indices give rise to binomid sequences (also known as Raney sequences) and generalized binomial coefficients. A finite binomid index $η$ can be extended to a unique lexicographically minimal infinite binomid index $\tildeη$. This lex-minimal extension $\tildeη$ is necessarily eventually periodic. In this research, we give a formula for the minimal period and provide an upper bound for the preperiod of $\tildeη$. We also show that the monoid of lex-minimal extensions is an inductive limit of finitely presented monoids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_20018 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the monoid of lexicographically minimal extensions Caalim, Jonathan Tanaka, Yu-ichi Combinatorics 11B65, 05A10, 06F05 A sequence $(e_i)_{i \le m}$ of nonnegative integers $e_i$, where $m \in \mathbb{N}$ or $m =\infty$, is called a binomid index if $\sum_{i=n-k+1}^{n} e_i\geq \sum_{i=1}^ke_i$ for all $k, n \in \mathbb{N}$ such that $ 1\le k \le n < m$. Infinite binomid indices give rise to binomid sequences (also known as Raney sequences) and generalized binomial coefficients. A finite binomid index $η$ can be extended to a unique lexicographically minimal infinite binomid index $\tildeη$. This lex-minimal extension $\tildeη$ is necessarily eventually periodic. In this research, we give a formula for the minimal period and provide an upper bound for the preperiod of $\tildeη$. We also show that the monoid of lex-minimal extensions is an inductive limit of finitely presented monoids. |
| title | On the monoid of lexicographically minimal extensions |
| topic | Combinatorics 11B65, 05A10, 06F05 |
| url | https://arxiv.org/abs/2505.20018 |