Related papers: A partition identity and the universal mock theta …

Andrews and Merca [J. Combin. Theory Ser. A 203 (2024), Art. 105849] recently obtained two interesting results on the sum of the parts with the same parity in the partitions of $n$ (the modulo $2$ case), the proof of which relies on…

We study certain bijection between plane partitions and $\mathbb{N}$-matrices. As applications, we prove a Cauchy-type identity for generalized dual Grothendieck polynomials. We introduce two statistics on plane partitions, whose generating…

The partition function $p(n)$, which counts the number of partitions of a positive integer $n$, is widely studied. Here, we study partition functions $p_S(n)$ that count partitions of $n$ into distinct parts satisfying certain congruence…

Recently Andrews and Bachraoui proved identities relating certain restricted partitions into distinct even parts with restricted 4-regular partitions by the theory of basic hypergeometric series. They also posed a question regarding…

In this paper we introduce k-run overpartitions as natural analogs to partitions without k-sequences, which were first defined and studied by Holroyd, Liggett, and Romik. Following their work as well as that of Andrews, we prove a number of…

Properties of four quintic theta functions are developed in parallel with those of the classical Jacobi null theta functions. The quintic theta functions are shown to satisfy analogues of Jacobi's quartic theta function identity and…

Let $f(q)$ denote Ramanujan's mock theta function \[f(q) = \sum_{n=0}^{\infty} a(n) q^{n} := 1+\sum_{n=1}^{\infty} \frac{q^{n^{2}}}{(1+q)^{2}(1+q^{2})^{2}\cdots(1+q^{n})^{2}}.\] It is known that there are many linear congruences for the…

The partition perimeter is a statistic defined to be one less than the sum of the number of parts and the largest part. Recently, Amdeberhan, Andrews, and Ballantine proved the following analog of Glaisher's theorem: for all $m \geq 2$ and…

Presented are polynomial identities which imply generalizations of Euler and Rogers--Ramanujan identities. Both sides of the identities can be interpreted as generating functions of certain restricted partitions. We prove the identities by…

In this short note, we prove several new congruences for the overcubic partition triples function, using both elementary techniques and the theory of modular forms. These extend the recent list of such congruences given by Nayaka,…

We find a close correspondence between certain partition functions of ideal quantum gases and certain symmetric polynomials. Due to this correspondence it can be shown that a number of thermodynamic identities which have recently been…

Based on a bijection due to Fu and Tang, we provide combinatorial proofs of several partition identities of Andrews and Merca. We also introduce two weights for partitions to extend one of these identities.

In a recent work, Andrews defined the singular overpartitions with the goal of presenting an overpartition analogue to the theorems of Rogers--Ramanujan type for ordinary partitions with restricted successive ranks. As a small part of his…

Motivated by Andrews' recent work related to Euler's partition theorem, we consider the set of partitions of an integer $n$ where the set of even parts has exactly $j$ elements, versus the set of partitions of $n$ where the set of repeated…

We study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as "sequential congruence": the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part…

In this paper, we use the Lambert series generating function for Euler's totient function to introduce a new identity for the number of $1$'s in the partitions of $n$. A new expansion for Euler's partition function $p(n)$ is derived in this…

We propose and recursively prove polynomial identities which imply Capparelli's partition theorems. We also find perfect companions to the results of Andrews, and Alladi, Andrews and Gordon involving $q$-trinomial coefficients. We follow…

In a previous paper of the second author with K. Ono, surprising multiplicative properties of the partition function were presented. Here, we deal with $k$-regular partitions. Extending the generating function for $k$-regular partitions…

In this note, we provide three new, very short proofs of two interesting congruences for Merca's partition function $a(n)$, which enumerates integer partitions where the odd parts have multiplicity at most 2. These modulo 2 congruences were…

For a positive integer $t$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello established infinite families of congruences and self-similarity results modulo $2$…