Counting Unions of Schreier Sets
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912124905717760 |
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| author | Beanland, Kevin Gorovoy, Dmitriy Hodor, Jȩdrzej Homza, Daniil |
| author_facet | Beanland, Kevin Gorovoy, Dmitriy Hodor, Jȩdrzej Homza, Daniil |
| contents | A subset of positive integers $F$ is a Schreier set if it is non-empty and $|F|\leqslant \min F$ (here $|F|$ is the cardinality of $F$). For each positive integer $k$, we define $k\mathcal{S}$ as the collection of all the unions of at most $k$ Schreier sets. Also, for each positive integer $n$, let $(k\mathcal{S})^n$ be the collection of all sets in $k\mathcal{S}$ with the maximum element equal to $n$. It is well-known that the sequence $(|(1\mathcal{S})^n|)_{n=1}^\infty$ is the Fibbonacci sequence. In particular, the sequence satisfies a linear recurrence. We generalize this statement, namely, we show that the sequence $(|(k\mathcal{S})^n|)_{n=1}^\infty$ satisfies a linear recurrence for every positive $k$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_01049 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Counting Unions of Schreier Sets Beanland, Kevin Gorovoy, Dmitriy Hodor, Jȩdrzej Homza, Daniil Combinatorics 05A19 A subset of positive integers $F$ is a Schreier set if it is non-empty and $|F|\leqslant \min F$ (here $|F|$ is the cardinality of $F$). For each positive integer $k$, we define $k\mathcal{S}$ as the collection of all the unions of at most $k$ Schreier sets. Also, for each positive integer $n$, let $(k\mathcal{S})^n$ be the collection of all sets in $k\mathcal{S}$ with the maximum element equal to $n$. It is well-known that the sequence $(|(1\mathcal{S})^n|)_{n=1}^\infty$ is the Fibbonacci sequence. In particular, the sequence satisfies a linear recurrence. We generalize this statement, namely, we show that the sequence $(|(k\mathcal{S})^n|)_{n=1}^\infty$ satisfies a linear recurrence for every positive $k$. |
| title | Counting Unions of Schreier Sets |
| topic | Combinatorics 05A19 |
| url | https://arxiv.org/abs/2211.01049 |