Realizability of Some Combinatorial Sequences
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916141801144320 |
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| author | Zhang, Geng-Rui |
| author_facet | Zhang, Geng-Rui |
| contents | A sequence $a=(a_n)_{n=1}^\infty$ of non-negative integers is called realizable if there is a self-map $T:X\to X$ on a set $X$ such that $a_n$ is equal to the number of periodic points of $T$ in $X$ of (not necessarily exact) period $n$, for all $n\geq1$. The sequence $a$ is called almost realizable if there exists a positive integer $m$ such that $(ma_n)_{n=1}^\infty$ is realizable. In this article, we show that certain wide classes of integer sequences are realizable, which contain many famous combinatorial sequences, such as the sequences of Apéry numbers of both kinds, central Delannoy numbers, Franel numbers, Domb numbers, Zagier numbers, and central trinomial coefficients. We also show that the sequences of Catalan numbers, Motzkin numbers, and large and small Schröder numbers are not almost realizable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_09454 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Realizability of Some Combinatorial Sequences Zhang, Geng-Rui Number Theory Combinatorics Dynamical Systems Primary 11B50, Secondary 37P35, 05A10, 11B83 A sequence $a=(a_n)_{n=1}^\infty$ of non-negative integers is called realizable if there is a self-map $T:X\to X$ on a set $X$ such that $a_n$ is equal to the number of periodic points of $T$ in $X$ of (not necessarily exact) period $n$, for all $n\geq1$. The sequence $a$ is called almost realizable if there exists a positive integer $m$ such that $(ma_n)_{n=1}^\infty$ is realizable. In this article, we show that certain wide classes of integer sequences are realizable, which contain many famous combinatorial sequences, such as the sequences of Apéry numbers of both kinds, central Delannoy numbers, Franel numbers, Domb numbers, Zagier numbers, and central trinomial coefficients. We also show that the sequences of Catalan numbers, Motzkin numbers, and large and small Schröder numbers are not almost realizable. |
| title | Realizability of Some Combinatorial Sequences |
| topic | Number Theory Combinatorics Dynamical Systems Primary 11B50, Secondary 37P35, 05A10, 11B83 |
| url | https://arxiv.org/abs/2302.09454 |