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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.

Recently, Brietzke, Silva and Sellers [‘Congruences related to an eighth order mock theta function of Gordon and McIntosh’, J. Math. Anal. Appl.479 (2019), 62–89] studied the number $v_{0}(n)$ of overpartitions of $n$ into odd parts without gaps between the nonoverlined parts, whose generating function is related to the mock theta function $V_{0}(q)$ of order 8. In this paper we first present a short proof of the 3-dissection for the generating function of $v_{0}(2n)$. Then we establish three congruences for $v_{0}(n)$ along certain progressions which are subsequences of the integers $4n+3$.

Let $\overline{t}(n)$ be the number of overpartitions in which (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd then it is overlined. Ramanujan-type congruences for $\overline{t}(n)$ modulo small powers of $2$ and $3$ have been established. We present two infinite families of congruences modulo $5$ and $27$ for $\overline{t}(n)$, the first of which generalises a recent result of Chern and Hao [‘Congruences for two restricted overpartitions’, Proc. Math. Sci. 129 (2019), Article 31].

We generalise a result of Chern [‘A curious identity and its applications to partitions with bounded part differences’, New Zealand J. Math. 47 (2017), 23–26] on distinct partitions with bounded difference between largest and smallest parts. The generalisation is proved both analytically and bijectively.

For positive integers $t_{1},\ldots ,t_{k}$, let $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ (respectively $p(n,t_{1},t_{2},\ldots ,t_{k})$) be the number of partitions of $n$ such that, if $m$ is the smallest part, then each of $m+t_{1},m+t_{1}+t_{2},\ldots ,m+t_{1}+t_{2}+\cdots +t_{k-1}$ appears as a part and the largest part is at most (respectively equal to) $m+t_{1}+t_{2}+\cdots +t_{k}$. Andrews et al. [‘Partitions with fixed differences between largest and smallest parts’, Proc. Amer. Math. Soc.143 (2015), 4283–4289] found an explicit formula for the generating function of $p(n,t_{1},t_{2},\ldots ,t_{k})$. We establish a $q$-series identity from which the formulae for the generating functions of $\tilde{p}(n,t_{1},t_{2},\ldots ,t_{k})$ and $p(n,t_{1},t_{2},\ldots ,t_{k})$ can be obtained.

A graph \begin{document}$G = (C, I, E)$\end{document} is called a split graph if its vertex set \begin{document}$V$\end{document} can be partitioned into a clique \begin{document}$C$\end{document} and an independent set \begin{document}$I$\end{document} . A graph \begin{document}$G = (Y \cup Z, I, E)$\end{document} is called a bisplit graph if its vertex set \begin{document}$V$\end{document} can be partitioned into three stable sets \begin{document}$I, Y,Z$\end{document} such that \begin{document}$Y \cup Z$\end{document} induces a complete bipartite graph and an independent set \begin{document}$I$\end{document} . A connected graph \begin{document}$G$\end{document} is called supper- \begin{document}$κ$\end{document} (resp. super- \begin{document}$λ$\end{document} ) if every minimum vertex cut (edge cut) of \begin{document}$G$\end{document} is the set of neighbors of some vertex (the edges of incident to some vertex) in \begin{document}$G$\end{document} . In this note, we show that: split graphs and bisplit graphs are super- \begin{document}$κ$\end{document} and super- \begin{document}$λ$\end{document} .

We are concerned with two types of partitions considered by Andrews, Lewis and Lovejoy. One is the partitions with designated summands where exactly one is tagged among parts with equal size. The other is the partitions with designated summands where all parts are odd. In this paper, we study two partition functions P D t ( n ) and P D O t ( n ) , which count the number of tagged parts over the above two types of partitions respectively. We first give the generating functions of P D t ( n ) and P D O t ( n ) . Then we establish many congruences modulo small powers of 3 for them. Finally, we pose some problems for future work.

Recently, Beck studied a new partition statistic which involves counting the total number of parts of a partition with certain rank or crank. Andrews proved two of Beck's conjectures related to ranks. Chern subsequently proved several results involving weighted rank and crank moments and deduced a number of similar Andrews-Beck type congruences. In this paper, we show that some of Chern's results can be explained by a simple combinatorial argument, and extend this approach to the study of k-colored partitions. As a consequence, we derive a large number of new Andrews-Beck type congruences for k-colored partitions.

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. Recently, congruences modulo powers of 2 for \(\overline{p}(n)\) were widely studied. In this paper, we prove several new infinite families of congruences modulo powers of 2 for \(\overline{p}(n)\). For example, for \(\alpha \ge 1\) and \(n\ge 0\), $$\begin{aligned} \overline{p}(8\cdot 3^{4\alpha +4}n+5\cdot 3^{4\alpha +3})\equiv 0 \quad (\mathrm{mod}\,\,{2^8}). \end{aligned}$$

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.

Recently, partitions with fixed or bounded differences between largest and smallest parts have attracted a lot of attention. In this paper, we first give a simple combinatorial proof of Breuer and Kronholm’s identity. Inspired by it, we construct a useful bijection to produce refinements of the results for partitions and overpartitions with bounded differences between largest and smallest parts. Consequently, we obtain Chern’s curious identity in a combinatorial manner.

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 this paper, we investigate the arithmetic properties of bipartitions with 3-core. Let A 3 ( n ) denote the number of bipartitions with 3-core of n. We will prove one infinite family of congruences modulo 5 for A 3 ( n ) . We also establish one surprising congruence modulo 14 for A 3 ( 8 n + 6 ) . Finally, we prove that, if u ( n ) denotes the number of representations of a nonnegative integer n in the form x 2 + y 2 + 3 z 2 + 3 t 2 with x , y , z , t ∈ Z , then u ( 6 n + 5 ) = 12 A 3 ( 2 n + 1 ) .

Recently, Keith used the theory of modular forms to study 9-regular partitions modulo 2 and 3. He obtained one infinite family of congruences modulo 3, and meanwhile proposed an analogous conjecture. In this note, we show that 9-regular partitions and 3-cores satisfy the same congruences modulo 3. Thus, we first derive several results on 3-cores, and then generalise Keith’s conjecture and get a stronger result, which implies that all of Keith’s results on congruences modulo 3 are consequences of our result.

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.

Recently, Radu and Sellers studied the parity of the function Δ 3 ( n ) which counts the number of broken 3-diamond partitions. They found interesting relationships between the generating functions in question and certain eta products. In this note, we aim to present elementary proofs of these congruence relationships.

In this paper, we study 2-, 3-, 4-, 6- and 12-dissections of a continued fraction of order twelve. Our main result is that when the aforementioned continued fraction and its reciprocal are expanded as power series, the sign of the coefficients is periodic, with period 12.

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 work, we consider various arithmetic properties of the function ped−2(n) that denotes the number of bipartitions of n with even parts distinct. We prove two infinite families of congruences for p−2(n) modulo 3. We also give characterizations of ped−2(n) modulo 2 and 4. Furthermore, for a fixed positive integer k, we show that ped−2(n) is divisible by 2k for almost all n.

Bringmann and Lovejoy introduced a rank for overpartition pairs and investigated its role in congruence properties of $\bar{pp}(n)$, the number of overpartition pairs of n. In particular, they applied the theory of Klein forms to show that there exist many Ramanujan-type congruences for the number $\bar{pp}(n)$. In this paper, we shall derive two Ramanujan-type identities and some explicit congruences for $\bar{pp}(n)$. Moreover, we find three ranks as combinatorial interpretations of the fact that $\bar{pp}(n)$ is divisible by three for any n. We also construct infinite families of congruences for $\bar{pp}(n)$ modulo 3, 5, and 9., 19 pages

Hirschhorn and Sellers studied arithmetic properties of the number of partitions with odd parts distinct. In another direction, Hammond and Lewis investigated arithmetic properties of the number of bipartitions. In this paper, we consider the number of bipartitions with odd parts distinct. Let this number be denoted by $pod_{-2}(n)$. We obtain two Ramanujan type identities for $pod_{-2}(n)$, which imply that $pod_{-2}(2n+1)$ is even and $pod_{-2}(3n+2)$ is divisible by 3. Furthermore, we show that for any $\alpha\geq 1$ and $n\geq 0$, $ pod_{-2}(3^{2\alpha+1}n+\frac{23\times 3^{2\alpha}-7}{8})$ is a multiple of 3 and $pod_{-2}(5^{\alpha+1}n+\frac{11\times 5^\alpha+1}{4})$ is divisible by 5. We also find combinatorial interpretations for the two congruences modulo 2 and 3., Comment: 15 pages

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.