Combinatorial proofs of some identities on overpartitions with repeated smallest non-overlined part
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911412877524992 |
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| author | Baruah, Nayandeep Deka Li, Haijun Mahanta, Pankaj Jyoti |
| author_facet | Baruah, Nayandeep Deka Li, Haijun Mahanta, Pankaj Jyoti |
| contents | Let $\overline{\mathrm{spt}}k(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part, say $s(π)$, appears $k$ times and every overlined part is bigger than $s(π)$. Let $\overline{\mathrm{spt}}k_o(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part appears $k$ times, every overlined part is bigger than $s(π)$ and all parts other than $s(π)$ are incongruent modulo $2$ with $s(π)$. Also, let $b_e(k,n)$ (resp., $b_o(k,n)$) denote the number of overpartitions of $n$ counted by $\overline{\mathrm{spt}}k_o(n)$ where the number of parts greater than $s(π)$ is even (resp., odd), and let $$\overline{\mathrm{spt}}k_o'(n)=b_e(k,n)-b_o(k,n).$$ Recently, Malik and Sarma (arXiv:2601.15601v1) expressed the generating functions of these partition functions in terms of linear combinations of $q$-series with polynomials in $q$ as coefficients. As corollaries, they derived some partition identities involving the functions for $k=1$ and sought for combinatorial proofs of their results. In this paper, we present some desired proofs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19736 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Combinatorial proofs of some identities on overpartitions with repeated smallest non-overlined part Baruah, Nayandeep Deka Li, Haijun Mahanta, Pankaj Jyoti Combinatorics 05A17, 05A15, 11P81 Let $\overline{\mathrm{spt}}k(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part, say $s(π)$, appears $k$ times and every overlined part is bigger than $s(π)$. Let $\overline{\mathrm{spt}}k_o(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part appears $k$ times, every overlined part is bigger than $s(π)$ and all parts other than $s(π)$ are incongruent modulo $2$ with $s(π)$. Also, let $b_e(k,n)$ (resp., $b_o(k,n)$) denote the number of overpartitions of $n$ counted by $\overline{\mathrm{spt}}k_o(n)$ where the number of parts greater than $s(π)$ is even (resp., odd), and let $$\overline{\mathrm{spt}}k_o'(n)=b_e(k,n)-b_o(k,n).$$ Recently, Malik and Sarma (arXiv:2601.15601v1) expressed the generating functions of these partition functions in terms of linear combinations of $q$-series with polynomials in $q$ as coefficients. As corollaries, they derived some partition identities involving the functions for $k=1$ and sought for combinatorial proofs of their results. In this paper, we present some desired proofs. |
| title | Combinatorial proofs of some identities on overpartitions with repeated smallest non-overlined part |
| topic | Combinatorics 05A17, 05A15, 11P81 |
| url | https://arxiv.org/abs/2601.19736 |