The heights of symmetric peaks and the depth of symmetric valleys over compositions of an integer
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913470965874688 |
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| author | Kezil, Walaa Asakly Noor |
| author_facet | Kezil, Walaa Asakly Noor |
| contents | A composition $π=π_1π_2\cdotsπ_k$ of a positive integer $n$ is an ordered collection of one or more positive integers whose sum is $n$ . The number of summands, namely $k$, is called the number of parts of $π$. In this paper, we introduce two statistics over compositions of an integer $n$ with exactly $k$ parts: heights of symmetric peaks and depths of symmetric valleys over all compositions of $n$. We derive an explicit formula for the generating functions of compositions of $n$ with exactly $k$ parts according to the number of symmetric peaks (valleys) and the total heights (depths) of peaks (valleys). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09166 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The heights of symmetric peaks and the depth of symmetric valleys over compositions of an integer Kezil, Walaa Asakly Noor Combinatorics A composition $π=π_1π_2\cdotsπ_k$ of a positive integer $n$ is an ordered collection of one or more positive integers whose sum is $n$ . The number of summands, namely $k$, is called the number of parts of $π$. In this paper, we introduce two statistics over compositions of an integer $n$ with exactly $k$ parts: heights of symmetric peaks and depths of symmetric valleys over all compositions of $n$. We derive an explicit formula for the generating functions of compositions of $n$ with exactly $k$ parts according to the number of symmetric peaks (valleys) and the total heights (depths) of peaks (valleys). |
| title | The heights of symmetric peaks and the depth of symmetric valleys over compositions of an integer |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2408.09166 |