43 problems

Let $\overline{p}(n)$ denote the number of overpartitions of $n$. For $n,k\geq 1$, Guo–Zeng's conjecture. … with strict inequality if $n\geq k^2$. The conjecture was proved indepen…

For a positive integer $n$, let $\overline{p}(n)$ denote the number of overpartitions of $n$. A Ramanujan-type congruence is a congruence of the form…

Bressoud's conjecture.

Let $\overline{OPT}_k(n)$ denote the number of overpartition $k$-tuples of $n$ with odd parts. Das–Saikia–Sarma's second conjecture. For all $i\geq 1$, all $n\geq 0$, and integers…

Let $\overline{p}_{q}(n)$ denote the number of overpartition $q$-tuples of $n$, where $q$ is prime. Saikia's conjecture. For all $n\geq 0$ and primes $q$, … These congruences exten…

Let $h(q)$ be the $q$-series … and let $(q)_\infty:=\prod_{n=1}^{\infty}(1-q^n)$. Andrews–Chan–Kim–Osburn's positivity conjecture. For every integer $m\geq 3$, the power series … h…

Let $\overline{p}_{9}(n)$ denote the $9$-color overpartition function, and let $n\ge0$ and $\alpha\ge1$. Arithmetic congruence conjecture. For all $n\ge0$ and $\alpha\ge1$, … These…

Let $\overline{p}_{6}(n)$ and $\overline{p}_{8}(n)$ denote the $2$-color overpartition functions with indicated color parameters, and let $n\ge0$. Higher-modulus congruence conject…

Let $\overline{p}_{\ell}(n)$ denote the $2$-color overpartition function, where $\ell$ is a positive integer, and let $n\ge0$. The mod 8 congruence conjecture. For all…

Let $\overline{P}(n,m)$ denote the number of overpartitions of $n$ with $m$ missing integers. Define $\overline{\mathcal{M}}_e(n)$ and $\overline{\mathcal{M}}_o(n)$ to be the numbe…

Let an overcolored partition of $n$ be a partition in which even parts may appear in one of $r$ colors and odd parts may appear in one of $s$ colors, with the first occurrence of e…

For a positive integer $n$, let $\overline{p}(n)$ denote the number of overpartitions of $n$, and let $\left(\frac{n}{p}\right)$ denote the Legendre symbol for an odd prime $p$. Co…

Let $\overline{p}(n)$ denote the overpartition function, and define $M_{\overline{p},k}(d)$ analogously to $M_k(d)$ as the largest index whose overpartition number lies within $d$…

Let $x_1,\dots,x_k,m,n\in\mathbb{N}$ and $j\in\{1,\dots,k\}$. Define $\overline{S_j}(x_1,\dots,x_k;m,n)$ as the colored overpartition count from Definition, with the fourth conditi…

Let $\overline{a}(n)$ denote the cubic overpartition function, and define its Jensen polynomials by … For each degree $d$, let $N_{\overline{a}}(d)$ be the minimum integer such tha…

Let $\overline{OPT}_k(n)$ denote the number of overpartition $k$-tuples of $n$ with odd parts. Das–Saikia–Sarma's first conjecture. For all $i,j\geq 1$, all $n\geq 0$, and positive…

Let $\overline{OPT}_k(n)$ denote the number of overpartition $k$-tuples of $n$ with odd parts. The four-congruence conjecture. For all $i\geq 1$, $n\geq 0$, and odd $r>0$, … These…

Let $\overline{M}_k(n)$ denote the number of overpartitions of $n$ in which the first part larger than $k$ appears at least $k+1$ times. For $n,k\geq 1$, Ballantine–Merca's conject…

Let $\overline{R_6^*}(n)$ denote the number of overpartitions of $n$ in which non-overlined parts are not divisible by $6$. Nath–Saikia–Sellers' conjecture. The function…

Let $\overline{R_{4,9}}(n)$ denote the number of $(4,9)$-regular overpartitions of $n$. For integers $n\geq0$, $\ell\geq2$, and $1\leq k\leq\ell$, Alanazi–Munagi–Saikia's conjectur…

Overcubic partition congruence conjecture modulo powers of three.

Let $\overline{R_6^\ast}(n)$ denote the overpartition function in which the nonoverlined parts are $6$-regular. Modulo 128 congruence family. For all $n \geq 0$ and $k \geq 1$, we…

Let $p_k(n)$ and $\overline{p}_k(n)$ denote the $k$-regular partition and $k$-regular overpartition functions, respectively, and let the inequalities referred to in the source as (…

Let $\overline{OPT}_k(n)$ denote the number of overpartition $k$-tuples with odd parts of size $n$. Let $i\geq 1$ and let $r$ be odd, with $n\geq 0$ and $n,i\in\mathbb{Z}$. Conject…

For odd $k$ among $5,7,9,13,15,17,21$, let $\overline{C}_{4k,k}(n)$ denote the corresponding $(4k,k)$-singular overpartition function. Parity-density conjecture. The values…