Distributions of Inversions and Descents over Integer Compositions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866910237179510784 |
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| author | Santos, E. G. |
| author_facet | Santos, E. G. |
| contents | We derive a generating function for the number of integer compositions of $n$ into $k$ parts (i.e., $k$-compositions of $n$) with a given number of inversions, and obtain similar results for $k$-compositions of $n$ with a given number of descents. Our approach relies on a known bijection that associates each integer composition $σ$ with a pair $(π,λ)$, where $π$ is a permutation and $λ$ is an integer partition. We show that the distribution of inversions and the distribution of descents over $k$-compositions are related, respectively, to the distribution of (maj,inv) and to the distribution of (inv,des) over permutations of $\{1,2,\ldots,k\}$, where maj, inv, and des denote the classical permutation statistics major index, inversion number, and descent number, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20253 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Distributions of Inversions and Descents over Integer Compositions Santos, E. G. General Mathematics 05A15, 05A05, 05A17 We derive a generating function for the number of integer compositions of $n$ into $k$ parts (i.e., $k$-compositions of $n$) with a given number of inversions, and obtain similar results for $k$-compositions of $n$ with a given number of descents. Our approach relies on a known bijection that associates each integer composition $σ$ with a pair $(π,λ)$, where $π$ is a permutation and $λ$ is an integer partition. We show that the distribution of inversions and the distribution of descents over $k$-compositions are related, respectively, to the distribution of (maj,inv) and to the distribution of (inv,des) over permutations of $\{1,2,\ldots,k\}$, where maj, inv, and des denote the classical permutation statistics major index, inversion number, and descent number, respectively. |
| title | Distributions of Inversions and Descents over Integer Compositions |
| topic | General Mathematics 05A15, 05A05, 05A17 |
| url | https://arxiv.org/abs/2605.20253 |