$s$-Modular, $s$-congruent and $s$-duplicate partitions
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| Format: | Preprint |
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2024
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| author | Nadji, Mohammed L. Moussa, Ahmia |
| author_facet | Nadji, Mohammed L. Moussa, Ahmia |
| contents | In this paper, we investigate the combinatorial properties of three classes of integer partitions: (1) $s$-modular partitions, a class consisting of partitions into parts with a number of occurrences (i.e., multiplicity) congruent to $0$ or $1$ modulo $s$, (2) $s$-congruent partitions, which generalize Sellers' partitions into parts not congruent to $2$ modulo $4$, and (3) $s$-duplicate partitions, of which the partitions having distinct odd parts and enumerated by the function $\mypod(n)$ are a special case. In this vein, we generalize Alladi's series expansion for the product generating function of $\mypod(n)$ and show that Andrews' generalization of Göllnitz-Gordon identities coincides with the number of partitions into parts simultaneously $s$-congruent and $t$-distinct (parts appearing fewer than $t$ times). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_13589 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $s$-Modular, $s$-congruent and $s$-duplicate partitions Nadji, Mohammed L. Moussa, Ahmia Combinatorics 05A17, 11P81, 11P83 In this paper, we investigate the combinatorial properties of three classes of integer partitions: (1) $s$-modular partitions, a class consisting of partitions into parts with a number of occurrences (i.e., multiplicity) congruent to $0$ or $1$ modulo $s$, (2) $s$-congruent partitions, which generalize Sellers' partitions into parts not congruent to $2$ modulo $4$, and (3) $s$-duplicate partitions, of which the partitions having distinct odd parts and enumerated by the function $\mypod(n)$ are a special case. In this vein, we generalize Alladi's series expansion for the product generating function of $\mypod(n)$ and show that Andrews' generalization of Göllnitz-Gordon identities coincides with the number of partitions into parts simultaneously $s$-congruent and $t$-distinct (parts appearing fewer than $t$ times). |
| title | $s$-Modular, $s$-congruent and $s$-duplicate partitions |
| topic | Combinatorics 05A17, 11P81, 11P83 |
| url | https://arxiv.org/abs/2408.13589 |