Congruences modulo powers of 5 and 7 for the crank and rank parity functions and related mock theta-functions

The Ramanujan Journal · 2026-05-18 · 2 citations

Multiplicative congruences for Andrews’s even parts below odd parts function and related infinite products

Arabian Journal of Mathematics · 2025-09-25 · 1 citations

An infinite family of overpartition congruences mod powers of 2

The Ramanujan Journal · 2025-12-01

An infinite family of overpartition congruences mod powers of 2

ArXiv.org · 2025-10-23

We prove an infinite family of Hecke-like congruences for the overpartition function modulo powers of 2. Starting from a recent identity of Garvan and Morrow and iterating Atkin's $U_2$ operator, we determine lower bounds on the 2-adic valuations of the coefficients that arise at each step. Our approach yields new modular equations relating the Hauptmoduln $G_2$ on $Γ_0(2)$ and $G_8$ on $Γ_0(8)$, together with explicit $U_2$-action formulas.

A simple proof of the Atkin-O'Brien partition congruence conjecture for powers of 13

ArXiv.org · 2025-04-15

In 1967, Atkin and O'Brien conjectured congruences for the partition function involving Hecke operators modulo powers of 13. In this paper, we provide a simple proof of this conjecture.

Multiplicative congruences for Andrews's even parts below odd parts function and related infinite products

ArXiv.org · 2025-05-02

We prove multiplicative congruences mod $2^{12}$ for George Andrews's partition function, $\overline{\mathcal{EO}}(n)$, the number of partitions of $n$ in which every even part is less than each odd part and only the largest even part occurs an odd number of times. We find analogous congruences for more general infinite products. These congruences are obtained using Fricke involutions and Newman's approach to half integer weight Hecke operators on eta quotients, and were inspired by Atkin's multiplicative congruences for the partition function.

Combinatorial Interpretations of Cranks of Overpartitions and Partitions without Repeated Odd Parts

arXiv (Cornell University) · 2024-06-17

We give combinatorial interpretations of two residual cranks of overpartitions defined by Bringmann, Lovejoy and Osburn in 2009 analogous to the crank of partitions given by Andrews and the first author in 1988. As a consequence, we give new versions of their definitions without adjusted weights. Furthermore, we investigate the combinatorial interpretation of an $M_2$-crank of partitions without repeated odd parts and explore connections of these statistics with their companion rank counterparts and the tenth order mock theta functions of Ramanujan.

Shifted and Shiftless Partition Identities

2024-07-03 · 1 citations

Let S and T be sets of positive integers. Let a be a fixed positive integer. A shifted partition identity has the form https://www.w3.org/1998/Math/MathML" display="block"> p ( S , n ) = p ( T , n − a ) , f o r a l l n ≥ a . https://www.w3.org/1999/xlink" content-type="black-white" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780138747060/455ed236-7e7e-4927-9fca-c6304fa53b43/content/math104_75_1_B.tif"/>

Combinatorial Interpretations of Cranks of Overpartitions and Partitions without Repeated Odd Parts

Symmetry Integrability and Geometry Methods and Applications · 2024-10-26

We give combinatorial interpretations of two residual cranks of overpartitions defined by Bringmann, Lovejoy and Osburn in 2009 analogous to the crank of partitions given by Andrews and the first author in 1988. As a consequence, we give new versions of their definitions without adjusted weights. Furthermore, we investigate the combinatorial interpretation of an $M_2$-crank of partitions without repeated odd parts and explore connections of these statistics with their companion rank counterparts and the tenth order mock theta functions of Ramanujan.

New symmetries for Dyson’s rank function

The Ramanujan Journal · 2024-04-03 · 2 citations