Linear Recurrences of Generalized Schreier Sets Revisited
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915797437251584 |
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| author | Chu, Hung Viet Vasseur, Zachary Louis |
| author_facet | Chu, Hung Viet Vasseur, Zachary Louis |
| contents | For $p, q\in \mathbb{N}$, a finite nonempty set $F$ is said to be $(p,q)$-Schreier (or maximal $(p,q)$-Schreier, respectively) if $q\min F\ge p|F|$ (or $q\min F = p|F|$, respectively). For $n\in \mathbb{N}$, let $$\mathcal{S}^{p/q}_{n}\ :=\ |\{F\subset\{1, 2, \ldots, n\}\,:\, q\min F\ge p|F|\mbox{ and }n\in F\}|.$$ Using the Inclusion-Exclusion Principle, Beanland et al. proved the recurrence $$|\mathcal{S}^{p/q}_{n}|\ =\ \sum_{k=1}^q(-1)^{k+1}\binom{q}{k}|\mathcal{S}^{p/q}_{n-k}| + |\mathcal{S}^{p/q}_{n-(p+q)}|.$$ We show that $(|\mathcal{S}^{p/q}_n|)_{n=1}^\infty$ is a subsequence with terms taken periodically from Padovan-like sequences which satisfy simple recurrence relations. As an application, we obtain an alternative proof of the above linear recurrence. Furthermore, a similar result holds for the sequence $(|\mathcal{M}^{p/q}_{n}|)_{n=1}^\infty$ that counts maximal $(p,q)$-Schreier sets. We end with a discussion of the relation between $(|\mathcal{S}^{p/q}_{n}|)_{n=1}^\infty$ and $(|\mathcal{M}^{p/q}_{n}|)_{n=1}^\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14312 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear Recurrences of Generalized Schreier Sets Revisited Chu, Hung Viet Vasseur, Zachary Louis Combinatorics Number Theory 05A19 (primary), 11B37, 11Y55, 05A15 (secondary) For $p, q\in \mathbb{N}$, a finite nonempty set $F$ is said to be $(p,q)$-Schreier (or maximal $(p,q)$-Schreier, respectively) if $q\min F\ge p|F|$ (or $q\min F = p|F|$, respectively). For $n\in \mathbb{N}$, let $$\mathcal{S}^{p/q}_{n}\ :=\ |\{F\subset\{1, 2, \ldots, n\}\,:\, q\min F\ge p|F|\mbox{ and }n\in F\}|.$$ Using the Inclusion-Exclusion Principle, Beanland et al. proved the recurrence $$|\mathcal{S}^{p/q}_{n}|\ =\ \sum_{k=1}^q(-1)^{k+1}\binom{q}{k}|\mathcal{S}^{p/q}_{n-k}| + |\mathcal{S}^{p/q}_{n-(p+q)}|.$$ We show that $(|\mathcal{S}^{p/q}_n|)_{n=1}^\infty$ is a subsequence with terms taken periodically from Padovan-like sequences which satisfy simple recurrence relations. As an application, we obtain an alternative proof of the above linear recurrence. Furthermore, a similar result holds for the sequence $(|\mathcal{M}^{p/q}_{n}|)_{n=1}^\infty$ that counts maximal $(p,q)$-Schreier sets. We end with a discussion of the relation between $(|\mathcal{S}^{p/q}_{n}|)_{n=1}^\infty$ and $(|\mathcal{M}^{p/q}_{n}|)_{n=1}^\infty$. |
| title | Linear Recurrences of Generalized Schreier Sets Revisited |
| topic | Combinatorics Number Theory 05A19 (primary), 11B37, 11Y55, 05A15 (secondary) |
| url | https://arxiv.org/abs/2506.14312 |