Related papers: Congruences satisfied by eta-quotients

To study statistical properties of modular forms, including for instance Sato-Tate like problems, it is essential to have a large number of Fourier coefficients. In this article, we exhibit three bases for the space of modular forms of any…

Congruences are found modulo powers of 5, 7 and 13 for Andrews' smallest parts partition function spt(n). These congruences are reminiscent of Ramanujan's partition congruences modulo powers of 5, 7 and 11. Recently, Ono proved explicit…

We prove some polynomial identities from which we deduce congruences modulo $p^2$ for the Fermat quotient $\frac{2^p-2}{p}$ for any odd prime $p$ (Proposition 1 and Theorem 1). These congruences are simpler than the one obtained by…

Lin introduced the partition function $\text{PDO}_t(n)$, which counts the total number of tagged parts over all the partitions of $n$ with designated summands in which all parts are odd. For $k\geq0$, Lin conjectured congruences for…

In 2015 Choi, Kim, and Lovejoy studied a weighted partition function, $A_1(m)$, which counted subpartitions with a structure related to the Rogers--Ramanujan identities. They conjectured the existence of an infinite class of congruences for…

In this paper, we derive certain congruences for the number of $3$-core cubic bipartitions using elementary $q$-series manipulations and dissection formulas.

For a positive integer $n$, let $B_{\ell_1,\dots,\ell_r}(n)$ denote the number of $(\ell_1,\ell_2,\cdots,\ell_r)$-regular multipartitions of $n$. If $\ell_1=\ell_2=\cdots=\ell_r=\ell$, then we denote $B_{\ell_1,\dots,\ell_r}(n)$ as…

For any positive integers $s$ and $t$, let $Q_{t}^{s}(n)$ denotes the number of partitions of a positive integer $n$ into distinct parts such that no part is congruent to $s$ or $t-s$ modulo $t$. We prove some Ramanujan-type congruences for…

Let $p_2(n)$ denote the number of cubic partitions. In this paper, we shall present two new congruences modulo $11$ for $p_2(n)$. We also provide an elementary alternative proof of a congruence established by Chan. Furthermore, we will…

An integral power series is called lacunary modulo $M$ if almost all of its coefficients are divisible by $M$. Motivated by the parity problem for the partition function, $p(n)$, Gordon and Ono studied the generating functions for…

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…

We provide an introduction of some basic facts of uniformly almost periodic functions, such as Fourier series representations. A result is then proved about Fourier coefficients which is a generalization of the purely periodic case. We then…

A partition of $n$ is $l$-regular if none of its parts is divisible by $l$. Let $b_l(n)$ denote the number of $l$-regular partitions of $n$. In this paper, using theta function identities due to Ramanujan, we establish some new infinite…

We present a generic partition refinement algorithm that quotients coalgebraic systems by behavioural equivalence, an important task in system analysis and verification. Coalgebraic generality allows us to cover not only classical…

The classical modular polynomials $\Phi_\ell(X,Y)$ give plane curve models for the modular curves $X_0(\ell)/\mathbb{Q}$ and have been extensively studied. In this article, we provide closed formulas for $\ell$ nontrivial coefficients of…

Traces of singular moduli were introduced and studied by Zagier in 1998. Being simultaneously the (traces of) values of a modular function ($j$-invariant) and Fourier coefficients of modular forms - which constitutes Zagier's duality -…

A new combinatorial object is introduced, the part-frequency matrix sequence of a partition, which is elementary to describe and is naturally motivated by Glaisher's bijection. We prove results that suggest surprising usefulness for such a…

In this article, we first investigate the partitions whose parts are congruent to $a$ or $b$ modulo $k$ with the aid of separable integer partition classes with modulus $k$ introduced by Andrews. Then, we introduce the…

Recently Lachterman, Schayer, and Younger published an elegant proof of the Ramanujan congruences for the partition function $p(n)$. Their proof uses only the classical theory of modular forms as well as a beautiful result of Choie, Kohnen,…

In this paper, we state and prove some congruence properties for the trinomial coeficients, one of which is similar to the Wolstenholme's theorem.