Related papers: Congruences in fractional partition functions

For rational $\alpha$, the fractional partition functions $p_\alpha(n)$ are given by the coefficients of the generating function $(q;q)^\alpha_\infty$. When $\alpha=-1$, one obtains the usual partition function. Congruences of the form…

The partition function $ p_{[1^c\ell^d]}(n)$ can be defined using the generating function, \[\sum_{n=0}^{\infty}p_{[1^c{\ell}^d]}(n)q^n=\prod_{n=1}^{\infty}\dfrac{1}{(1-q^n)^c(1-q^{\ell n})^d}.\] In \cite{P}, we proved infinite family of…

Let $p$ be prime, and let $p_{[1,p]}(n)$ denote the function whose generating function is $\prod (1-q^n)^{-1}(1 - q^{pn})^{-1}$. This function and its generalizations $p_{[c^{\ell}, d^m]}(n)$ are the subject of study in several recent…

Let $p_{k}(n)$ be the coefficient of $q^n$ in the series expansion of $(q;q)_{\infty}^{k}$. It is known that the partition function $p(n)$, which corresponds to the case when $k=-1$, satisfies congruences such as $p(5n+4)\equiv 0\pmod{5}$.…

The arithmetic properties of the ordinary partition function $p(n)$ have been the topic of intensive study for the past century. Ramanujan proved that there are linear congruences of the form $p(\ell n+\beta)\equiv 0\pmod\ell$ for the…

Recently, Chan and Wang (Fractional powers of the generating function for the partition function. Acta Arith. 187(1), 59--80 (2019)) studied the fractional powers of the generating function for the partition function and found several…

The partition function $ p_{[1^c11^d]}(n)$ can be defined using the generating function, \[\sum_{n=0}^{\infty}p_{[1^c{11}^d]}(n)q^n=\prod_{n=1}^{\infty}\dfrac{1}{(1-q^n)^c(1-q^{11 n})^d}.\] In this paper, we prove infinite families of…

We consider a class of generating functions analogous to the generating function of the partition function and establish a bound on the primes $\ell$ for which their coefficients $c(n)$ obey congruences of the form $c(\ell n + a) \equiv 0…

Let $p(n)$ denote the partition function. In this article, we will show that congruences of the form $$ p(m^j\ell^kn+B)\equiv 0\mod m \text{for all} n\ge 0 $$ exist for all primes $m$ and $\ell$ satisfying $m\ge 13$ and $\ell\neq 2,3,m$.…

In recent work with Raum the authors considered congruences for the ordinary partition function $p(n)$ of the form $p(\ell Q^r n+\beta)\equiv 0\pmod\ell$ where $\ell, Q\geq 5$ are prime and $r\in \{1,2\}$, and proved a number of results…

Let $p(n)$ be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form $p( Q^3 \ell n+\beta)\equiv0\pmod\ell$ where $\ell$ and $Q$ are prime and $5\leq \ell\leq 31$; these lie in two natural…

Let $p_{-t}(n)$ denote the number of partitions of $n$ into $t$ colors. In analogy with Ramanujan's work on the partition function, Lin recently proved in \cite{Lin} that $p_{-3}(11n+7)\equiv0\pmod{11}$ for every integer $n$. Such…

Let $\ell \geq 5$ be prime. For the partition function $p(n)$ and $5 \leq \ell \leq 31$, Atkin found a number of examples of primes $Q \geq 5$ such that there exist congruences of the form $p(\ell Q^{3} n+\beta) \equiv 0 \pmod{\ell}.$…

For a given prime $p$, we study the properties of the $p$-dissection identities of Ramanujan's theta functions $\psi(q)$ and $f(-q)$, respectively. Then as applications, we find many infinite family of congruences modulo 2 for some…

Let $p_k(n)$ be given by the $k$-th power of the Euler Product $\prod _{n=1}^{\infty}(1-q^n)^k=\sum_{n=0}^{\infty}p_k(n)q^{n}$. By investigating the properties of the modular equations of the second and the third order under the Atkin…

In the 1960s Atkin discovered congruences modulo primes $\ell\leq 31$ for the partition function $p(n)$ in arithmetic progressions modulo $\ell Q^3$, where $Q\neq \ell$ is prime. Recent work of the first author with Allen and Tang shows…

Let $p_r(n)$ denote the number of $r$-component multipartitions of $n$, and let $S_{\gamma,\lambda}$ be the space spanned by $\eta(24z)^\gamma \phi(24z)$, where $\eta(z)$ is the Dedekind's eta function and $\phi(z)$ is a holomorphic modular…

Folsom, Kent, and Ono used the theory of modular forms modulo $\ell$ to establish remarkable ``self-similarity'' properties of the partition function and give an overarching explanation of many partition congruences. We generalize their…

New congruences are found for Andrews' smallest parts partition function spt(n). The generating function for spt(n) is related to the holomorphic part alpha(24z) of a certain weak Maass form M(z) of weight 3/2. We show that a normalized…

Recently, Andrews, Dixit and Yee introduced partition functions associated with Ramanujan/Watson third order mock theta functions $\omega(q)$ and $\nu(q)$. In this paper, we find several new exact generating functions for those partition…