Study of $p^{k}$-Eulerian polynomials and $p^{k}$-Fibonacci numbers for every odd prime $p$ and $k\geq0$
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arXiv
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| Format: | Preprint |
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2025
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| author | Parvathi, M. Tamilselvi, A. Hepsi, D. |
| author_facet | Parvathi, M. Tamilselvi, A. Hepsi, D. |
| contents | In this paper, we define the notion of descent for the paths in the $p$-Bratteli diagram. This leads to the definition of $p^{k}$-Eulerian polynomials, whose coefficients count the number of paths with a given number of descents. We provide a method for constructing the $p^{k}$-Eulerian polynomials at each vertex. Furthermore, we compute the total number of descents of all paths ending at a given vertex as the corresponding $p^{k}$-Fibonacci numbers. We show that the derivative of the $p^{k}$-Eulerian polynomial evaluated at 1 for a fixed vertex equals the corresponding $p^{k}$-Fibonacci number. Finally, we discuss the generating function for the sequence of $p^{k}$-Fibonacci numbers and the recurrence relations they satisfy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_09941 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Study of $p^{k}$-Eulerian polynomials and $p^{k}$-Fibonacci numbers for every odd prime $p$ and $k\geq0$ Parvathi, M. Tamilselvi, A. Hepsi, D. Combinatorics primary 05E10, 05A15, secondary 05E16 In this paper, we define the notion of descent for the paths in the $p$-Bratteli diagram. This leads to the definition of $p^{k}$-Eulerian polynomials, whose coefficients count the number of paths with a given number of descents. We provide a method for constructing the $p^{k}$-Eulerian polynomials at each vertex. Furthermore, we compute the total number of descents of all paths ending at a given vertex as the corresponding $p^{k}$-Fibonacci numbers. We show that the derivative of the $p^{k}$-Eulerian polynomial evaluated at 1 for a fixed vertex equals the corresponding $p^{k}$-Fibonacci number. Finally, we discuss the generating function for the sequence of $p^{k}$-Fibonacci numbers and the recurrence relations they satisfy. |
| title | Study of $p^{k}$-Eulerian polynomials and $p^{k}$-Fibonacci numbers for every odd prime $p$ and $k\geq0$ |
| topic | Combinatorics primary 05E10, 05A15, secondary 05E16 |
| url | https://arxiv.org/abs/2506.09941 |