Related papers: Congruences for Restricted Plane Overpartitions Mo…

It is proved that the number of 9-regular partitions of n is divisible by 3 when n is congruent to 3 mod 4, and by 6 when n is congruent to 13 mod 16. An infinite family of congruences mod 3 holds in other progressions modulo powers of 4…

Let $\overline{B}_{s,t}(n)$ denote the number of overpartitions of $n$ where no part is divisible by $s$ or $t$, with $s$ and $t$ being coprime. By establishing the exact generating functions of a family of arithmetic progressions in…

In this article we exhibit new explicit families of congruences for the overpartition function, making effective the existence results given previously by Treneer. We give infinite families of congruences modulo $m$ for $m = 5, 7, 11$, and…

Alanzi et al. (2022) investigated overpartition of a positive integer $n$ with $\ell$-regular non-overlined parts denoted by $\overline R_\ell^\ast (n)$, and proved some results for the case $\ell=3$. As extension to the results of Alanzi…

In this paper we study restricted overpartitions and concave compositions. In several cases the resulting generating functions involve simultaneously modular forms, mock theta functions, mock Maass theta functions, and false theta…

The $(k,i)$-singular overpartitions, combinatorial objects introduced by Andrews in 2015, are known to satisfy Ramanujan-type congruences modulo any power of prime coprime to $6k$. In this paper we consider the parity of the number…

Recently, Lin introduced two new partition functions PD$_t(n)$ and PDO$_t(n)$, which count the total number of tagged parts over all partitions of $n$ with designated summands and the total number of tagged parts over all partitions of $n$…

We find and prove a class of congruences modulo 4 for Andrews' partition with certain ternary quadratic form. We also discuss distribution of $\overline{\mathcal{EO}}(n)$ and further prove that $\overline{\mathcal{EO}}(n)\equiv0\pmod4$ for…

Let ${{\overline{p}}_{3}}(n)$ be the number of overpartition triples of $n$. By elementary series manipulations, we establish some congruences for ${\overline{p}}_{3}(n)$ modulo small powers of 2, such as…

We improve on the lower bound of the maximum number of planes of ${\rm PG}(8,q)$ mutually intersecting in at most one point leading to the following lower bound: ${\cal A}_q(9, 4; 3) \ge q^{12}+2q^8+2q^7+q^6+q^5+q^4+1$ for constant…

By considering the $M_2$-rank of an overpartition as well as a residual crank, we give another combinatorial refinement of the congruences $\overline{\mbox{spt}}_2(3n)\equiv \overline{\mbox{spt}}_2(3n+1)\equiv 0\pmod{3}$. Here…

The starting point for this work is the family of functions $\overline{p}_{-t}(n)$ which counts the number of $t$--colored overpartitions of $n.$ In recent years, several infinite families of congruences satisfied by $\overline{p}_{-t}(n)$…

A classical result asserts that the complex projective plane modulo complex conjugation is the 4-dimensional sphere. We generalize this result in two directions by considering the projective planes over the normed real division algebras and…

In this paper, we study the partition functions $\overline{R_\ell^\ast}(n)$, which count the number of overpartitions of $n$ where the non-overlined parts are $\ell$-regular for a given $\ell$. Using elementary techniques, as well as the…

Let $\overline{bt}(n)$ denote the number of overcubic partition triples of $n$. Nayaka, Dharmendra and Kumar proved some congruences modulo 8, 16 and 32 for $\overline{bt}(n)$. Recently, Saikia and Sarma established some congruences modulo…

Let $\overline{p}_o(n)$ denote the number of overpartitions of $n$ into odd parts. The partition function $\overline{p}_o(n)$ has been the subject of many recent studies where many explicit Ramanujan-like congruences were discovered. In…

The partition function is known to exhibit beautiful congruences that are often proved using the theory of modular forms. In this paper, we study the extent to which these congruence results apply to the generalized Frobenius partitions…

Using elementary methods, we prove new formulas for $\operatorname{pp}(n)$, the number of plane partitions of $n$, $\operatorname{pp}_r(n)$, the number of plane partitions of $n$ with at most $r$ rows, $\operatorname{pp}^s(n)$, the number…

In this paper we give a full description of the inequalities that can occur between overpartition ranks. If $ \overline{N}(a,c,n) $ denotes the number of overpartitions of $ n $ with rank congruent to $ a $ modulo $ c,$ we prove that for…

Using the results of Dalla Piazza, Fiorentino and Salvati Manni, we determine an explicit modular form defining the locus of plane quartics with a hyperflex among all plane quartics. As a result, we provide a direct way to compute the…