Divisibility of an analogue of $t$-core partition function by powers of primes
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| Format: | Preprint |
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2024
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| author | Talukdar, Pranjal |
| author_facet | Talukdar, Pranjal |
| contents | A partition of a positive integer $n$ is said to be $t$-core if none of its hook lengths are divisible by $t$. Recently, two analogues, $\overline{a}_t(n)$ and $\overline{b}_t(n)$, of the $t$-core partition function, $c_t(n)$, have been introduced by Gireesh, Ray and Shivashankar \cite{grs} and Bandyopadhyay and Baruah \cite{bb}, respectively. In this article, we prove the lacunarity of $\overline{b}_t(n)$ modulo arbitrary powers of 2 and 3 for $t=3^αm$ where $\gcd(m,6)$=1. For a fixed positive integer $k$ and prime numbers $p_i\geq 5$, we also study the arithmetic density of $\overline{b}_t(n)$ modulo $p_i^k$ where $t=p_1^{a_1}\cdots p_m^{a_m}$. We further prove an infinite family of congruences for $\overline{b}_3(n)$ modulo arbitrary powers of 2 by employing a result of Ono and Taguchi on the nilpotency of Hecke operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_05274 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Divisibility of an analogue of $t$-core partition function by powers of primes Talukdar, Pranjal Number Theory 11P83, 05A17, 11F11 A partition of a positive integer $n$ is said to be $t$-core if none of its hook lengths are divisible by $t$. Recently, two analogues, $\overline{a}_t(n)$ and $\overline{b}_t(n)$, of the $t$-core partition function, $c_t(n)$, have been introduced by Gireesh, Ray and Shivashankar \cite{grs} and Bandyopadhyay and Baruah \cite{bb}, respectively. In this article, we prove the lacunarity of $\overline{b}_t(n)$ modulo arbitrary powers of 2 and 3 for $t=3^αm$ where $\gcd(m,6)$=1. For a fixed positive integer $k$ and prime numbers $p_i\geq 5$, we also study the arithmetic density of $\overline{b}_t(n)$ modulo $p_i^k$ where $t=p_1^{a_1}\cdots p_m^{a_m}$. We further prove an infinite family of congruences for $\overline{b}_3(n)$ modulo arbitrary powers of 2 by employing a result of Ono and Taguchi on the nilpotency of Hecke operators. |
| title | Divisibility of an analogue of $t$-core partition function by powers of primes |
| topic | Number Theory 11P83, 05A17, 11F11 |
| url | https://arxiv.org/abs/2405.05274 |