Weighted Schreier-type Sets and the Fibonacci Sequence
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910464010616832 |
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| author | Chu, Hung Viet Vasseur, Zachary Louis |
| author_facet | Chu, Hung Viet Vasseur, Zachary Louis |
| contents | For a finite set $A\subset\mathbb{N}$ and $k\in \mathbb{N}$, let $ω_k(A) = \sum_{i\in A, i\neq k}1$. For each $n\in \mathbb{N}$, define $$a_{k, n}\ =\ |\{E\subset \mathbb{N}\,:\, E = \emptyset\mbox{ or } ω_k(E) < \min E\leqslant \max E\leqslant n\}|.$$ First, we prove that $$a_{k,k+\ell} \ =\ 2F_{k+\ell},\mbox{ for all }\ell\geqslant 0\mbox{ and }k\geqslant \ell+2,$$ where $F_n$ is the $n$th Fibonacci number. Second, we show that $$|\{E\subset \mathbb{N}\,:\, \max E = n+1, \min E > ω_{2,3}(E), \mbox{ and }|E|\neq 2\}|\ =\ F_{n},$$ where $ω_{2,3}(E) = \sum_{i\in E, i\neq 2, 3}1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19352 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weighted Schreier-type Sets and the Fibonacci Sequence Chu, Hung Viet Vasseur, Zachary Louis Combinatorics Number Theory 11Y55, 11B37 For a finite set $A\subset\mathbb{N}$ and $k\in \mathbb{N}$, let $ω_k(A) = \sum_{i\in A, i\neq k}1$. For each $n\in \mathbb{N}$, define $$a_{k, n}\ =\ |\{E\subset \mathbb{N}\,:\, E = \emptyset\mbox{ or } ω_k(E) < \min E\leqslant \max E\leqslant n\}|.$$ First, we prove that $$a_{k,k+\ell} \ =\ 2F_{k+\ell},\mbox{ for all }\ell\geqslant 0\mbox{ and }k\geqslant \ell+2,$$ where $F_n$ is the $n$th Fibonacci number. Second, we show that $$|\{E\subset \mathbb{N}\,:\, \max E = n+1, \min E > ω_{2,3}(E), \mbox{ and }|E|\neq 2\}|\ =\ F_{n},$$ where $ω_{2,3}(E) = \sum_{i\in E, i\neq 2, 3}1$. |
| title | Weighted Schreier-type Sets and the Fibonacci Sequence |
| topic | Combinatorics Number Theory 11Y55, 11B37 |
| url | https://arxiv.org/abs/2405.19352 |