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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 $$p_{\omega }(n)$$ .

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.

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.

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 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, 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.

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

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.