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Let $$p_{\omega }(n)$$ be the number of partitions of n in which each odd part is less than twice the smallest part, and assume that $$p_{\omega }(0)=0$$ . Andrews, Dixit, and Yee proved that the generating function of $$p_{\omega }(n)$$ equals $$q\omega (q)$$ , where $$\omega (q)$$ is one of the third order mock theta functions. Many scholars have studied the arithmetic properties of $$p_{\omega }(n)$$ . For example, Waldherr proved that $$p_{\omega }(40n+28)\equiv p_{\omega }(40n+36)\equiv 0 \pmod 5$$ by using the theory of Maass forms, which was later proved once again in an elementary method by Andrews, Passary, Sellers, and Yee. In this paper, we shall establish four new congruences modulo 5 for $$p_{\omega }(n)$$ .

Stanley introduced a partition statistic $$srank (\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$$ , where $$\mathcal {O}(\pi )$$ denote the number of odd parts of the partition $$\pi $$ , and $$\pi '$$ is the conjugate of $$\pi $$ . Let $$p_i(n)$$ denote the number of partitions of n with srank $$\equiv i\pmod 4$$ . Andrews proved the following refinement of Ramanujan’s partition congruence modulo 5: $$\begin{aligned} p_0(5n+4)\equiv p_2(5n+4)\equiv 0\pmod 5. \end{aligned}$$ In this paper, we consider an analogous partition statistic $$\begin{aligned} lrank (\pi )=\mathcal {O}(\pi )+\mathcal {O}(\pi '). \end{aligned}$$ Let $$p_i^+(n)$$ denote the number of partitions of n with lrank $$\equiv i \pmod 4$$ . We will establish the generating functions of $$p_0^+(n)$$ and $$p_2^+(n)$$ and show that they satisfy similar properties to $$p_i(n)$$ . We also utilize a pair of interesting q-series identities to obtain a direct proof of the congruences $$\begin{aligned} p_0^+(5n+4)\equiv p_2^+(5n+4)\equiv 0\pmod 5. \end{aligned}$$

Recently, partitions with fixed or bounded difference between largest and smallest parts have attracted a lot of attention. In this paper, we provide both analytic and combinatorial proofs of the generating function for k-regular partitions with bounded difference kt between largest and smallest parts. Inspired by Franklin’s result, we further find a new proof of the generating function for overpartitions with bounded part differences by using Dousse and Kim’s results on (q, z)-overGaussian polynomials.

Let b(n) be the number of cubic partition pairs of n. We establish three congruences for b(n) modulo arbitrary powers of 3. Meanwhile, we prove some conjectures proposed by the first author.

Let b ( n ) denote the number of cubic partition pairs of n. This paper aims to study the congruences for b ( n ) modulo 27. We first establish three Ramanujan type congruences. Then many infinite families of congruences are presented. Finally, we propose two conjectures on the congruences for b ( n ) modulo 49 , 81 and 243.

Let $$b_{13}(n)$$ denote the number of partitions of n such that no parts are divisible by 13. In this paper, we shall prove several infinite families of congruence relations modulo 2 for $$b_{13}(n)$$ . In addition, we will give an elementary proof of the parity result on broken 6-diamond partitions, which was established by Radu and Sellers. We also find a new congruence relation modulo 2 for the number of broken 6-diamond partitions of n.

Let \(\overline{p}(n)\) denote the number of overpartitions of n. In this paper, we shall show that for \(n\ge 0\), $$\begin{aligned} \overline{p}(80n+r)\equiv 0\ (\mathrm{mod\ }5), \end{aligned}$$ where \(r=8,52,68\), and 72. In addition, we present a short alternative proof of the congruence $$\begin{aligned} \overline{p}(40n+35)\equiv 0\ (\mathrm{mod\ }5), \end{aligned}$$ which is conjectured by Hirschhorn and Sellers.

In this paper, we investigate congruences for [Formula: see text] modulo 2 and characterize the parity of [Formula: see text] and [Formula: see text] according to the arithmetic property of [Formula: see text]. As a consequence, we obtain various Ramanujan type congruences for [Formula: see text]. We also extend these results to several infinite families of congruences.

In this note, we establish two identities of \((q;\,q)_\infty ^8\) by using Jacobi’s four-square theorem and two of Ramanujan’s identities. As an important consequence, we present one Ramanujan-style proof of the congruence \(p_{-3}(11n+7)\equiv 0\ (\mathrm{mod\ }11)\), where \(p_{-3}(n)\) denotes the number of 3-color partitions of n.

Let \(B_{13}(n)\) denote the number of \(13\)-regular bipartitions of \(n\). Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function \(p(n)\). In particular, we shall prove an infinite family of congruences: for \(\alpha \ge 2\) and \(n\ge 0\), $$\begin{aligned} B_{13}(3^{\alpha }n+2\cdot 3^{\alpha -1}-1)\equiv \ 0\ (\mathrm{mod\ }3). \end{aligned}$$ In addition, we will also give an alternative proof of one infinite family of congruences for \(b_{13}(n)\), the number of \(13\) regular partitions of \(n\), due to Webb.

Let $$\overline{p}(n)$$ denote the number of overpartitions of $$n$$ . Hirschhorn and Sellers conjectured that for $$n\ge 0$$ , $$\overline{p}(40n + 35)\equiv 0\ (\mathrm{mod\ }40)$$ . Recently, Chen and Xia have proven this conjecture using 2-dissection formulas of theta functions due to Ramanujan, and Hirschhorn and Sellers, as well as $$(p,k)$$ -parametrization of theta functions given by Alaca, Alaca and Williams. In this paper, we shall give a new proof of this conjecture using some identities of Ramanujan. We also show that Ramanujan type identity for partitions weighted by the parity of the crank due to Choi, Kang and Lovejoy is equivalent with Ramanujan’s “Most Beautiful Identity” on $$p(5n+4)$$ , where $$p(n)$$ denotes the number of partitions of $$n$$ .

In 2007, Andrews and Paule introduced a new class of combinatorial objects called broken k-diamonds. Let Δk(n) be the number of broken k-diamonds partitions of n. By using MacMahon’s partition analysis, they found that the generating function of Δk(n) is an infinite product, which attracted many mathematicians to study its congruence properties. Recently, Radu and Sellers found infinitely many Ramanujan-like congruences modulo 3 for Δ2(n). In fact, they obtained 3-dissections of the generating function for Δ2(n) modulo 3 by using the theory of modular forms. In this brief note, we aim to present elementary proofs of Radu and Sellers’ results by using several theta function identities due to Ramanujan.

In this paper, we study the function B l (n) which counts the number of l-regular bipartitions of n. Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n). In particular, using Ramanujan’s two modular equations of degree 7, we prove an infinite family of congruences: for α≥2 and n≥0, $$B_7 \biggl(3^{\alpha}n+\frac{5\cdot 3^{\alpha-1}-1}{2} \biggr)\equiv 0\ ({ \rm mod\ }3). $$ In addition, we give an elementary proof of two infinite families of congruences modulo 3 satisfied by the 7-regular partition function due to Furcy and Penniston (Ramanujan J. 27:101–108, 2012). We also present two conjectures for B 13(n) modulo 3.

In this paper, we investigate the divisibility of the function b(n), counting the number of certain restricted 3-colored partitions of n. We obtain one Ramanujan type identity, which implies that b(3n + 2) ≡ 0 ( mod 3). Furthermore, we study the generating function for b(3n + 1) by modular forms. Finally, we find two cranks as combinatorial interpretations of the fact that b(3n + 2) is divisible by 3 for any n.

In this brief note, we prove one unexpected Ramanujan type congruence modulo 7 for the number cϕ4(n) of generalized Frobenius partitions of n with four colors.

Let $\overline{b}(n)$ denote the number of overcubic partition pairs of $n$. In this paper, we establish two Ramanujan type congruences and several infinite families of congruences modulo $3$ satisfied by $\overline{b}(n)$ . For modulus $5$, we obtain one Ramanujan type congruence and two congruence relations for $\overline{b}(n)$, from which some strange congruences are derived.