Further Results for the Capacity Statistic Distribution on Compositions of 1's and 2's
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917894803161088 |
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| author | Shattuck, Mark |
| author_facet | Shattuck, Mark |
| contents | In this paper, we study additional aspects of the capacity distribution on the set $\mathcal{B}_n$ of compositions of $n$ consisting of $1$'s and $2$'s. Among our results are further recurrences for this distribution as well as formulas for the total capacity and sign balance on $\mathcal{B}_n$. We provide algebraic and combinatorial proofs of our results. We also give combinatorial explanations of some prior results where such a proof was requested. Finally, the joint distribution of the capacity statistic with two further parameters on $\mathcal{B}_n$ is briefly considered. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_09931 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Further Results for the Capacity Statistic Distribution on Compositions of 1's and 2's Shattuck, Mark Combinatorics 05A05 In this paper, we study additional aspects of the capacity distribution on the set $\mathcal{B}_n$ of compositions of $n$ consisting of $1$'s and $2$'s. Among our results are further recurrences for this distribution as well as formulas for the total capacity and sign balance on $\mathcal{B}_n$. We provide algebraic and combinatorial proofs of our results. We also give combinatorial explanations of some prior results where such a proof was requested. Finally, the joint distribution of the capacity statistic with two further parameters on $\mathcal{B}_n$ is briefly considered. |
| title | Further Results for the Capacity Statistic Distribution on Compositions of 1's and 2's |
| topic | Combinatorics 05A05 |
| url | https://arxiv.org/abs/2501.09931 |