d-Fold Partition Diamonds
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929337876348928 |
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| author | Dockery, Dalen Jameson, Marie Sellers, James A. Wilson, Samuel |
| author_facet | Dockery, Dalen Jameson, Marie Sellers, James A. Wilson, Samuel |
| contents | In this work we introduce new combinatorial objects called $d$--fold partition diamonds, which generalize both the classical partition function and the partition diamonds of Andrews, Paule and Riese, and we set $r_d(n)$ to be their counting function. We also consider the Schmidt type $d$--fold partition diamonds, which have counting function $s_d(n).$ Using partition analysis, we then find the generating function for both, and connect the generating functions $\sum_{n= 0}^\infty s_d(n)q^n$ to Eulerian polynomials. This allows us to develop elementary proofs of infinitely many Ramanujan--like congruences satisfied by $s_d(n)$ for various values of $d$, including the following family: for all $d\geq 1$ and all $n\geq 0,$ $s_d(2n+1) \equiv 0 \pmod{2^d}.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_02579 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | d-Fold Partition Diamonds Dockery, Dalen Jameson, Marie Sellers, James A. Wilson, Samuel Number Theory Combinatorics 11P82 (primary), 11P83 (secondary) In this work we introduce new combinatorial objects called $d$--fold partition diamonds, which generalize both the classical partition function and the partition diamonds of Andrews, Paule and Riese, and we set $r_d(n)$ to be their counting function. We also consider the Schmidt type $d$--fold partition diamonds, which have counting function $s_d(n).$ Using partition analysis, we then find the generating function for both, and connect the generating functions $\sum_{n= 0}^\infty s_d(n)q^n$ to Eulerian polynomials. This allows us to develop elementary proofs of infinitely many Ramanujan--like congruences satisfied by $s_d(n)$ for various values of $d$, including the following family: for all $d\geq 1$ and all $n\geq 0,$ $s_d(2n+1) \equiv 0 \pmod{2^d}.$ |
| title | d-Fold Partition Diamonds |
| topic | Number Theory Combinatorics 11P82 (primary), 11P83 (secondary) |
| url | https://arxiv.org/abs/2307.02579 |