91 problems

Dyson's rank equidistribution conjecture. For all nonnegative integers $n$, one has

Let $d_k(n)$ be defined by the generating function … In particular, $d_2(n)$ counts the relevant 2-elongated plane partitions. Andrews–Paule conjecture. For all integers $n\geq 0$…

Andrews and Ghosh Dastidar's conjecture. Under this condition,

Churchhouse's conjecture. For every positive integer $n$,

Let $f_m:=\prod_{j=1}^{\infty}(1-q^{jm})$ for every positive integer $m$, and let $\text{PDO}_t(n)$ denote the total number of tagged parts over all partitions of $n$ with designat…

A partition with designated summands is obtained from an ordinary partition by tagging exactly one occurrence of each part size. Let $\text{PDO}_t(n)$ denote the total number of ta…

Let $\textup{spt}_\omega(n)$ denote the partition statistic used in the paper. For integers $k\geq 1$ and $n\geq 0$, Wang's conjecture. … … The conjecture concerns an infinite fami…

Mao's conjectured inequalities. Computation evidence suggests that

Let $n\geq 0$, $i\geq 3$, and $k\geq 0$. The overcubic partition tuple function $\overline{b}_m(r)$ is considered at the indicated arithmetic progressions. Congruence conjecture. F…

Banerjee–Bringmann–Bachraoui's conjecture. For all integers $n$ and $k$,

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…

Dyadic congruence conjecture. For every integer $n\geq 0$, the congruences

Let $DSOME(n)$ denote the sum of all odd parts in the partitions of $n$ into distinct parts minus the sum of all even parts. DSOME congruence conjecture. For all integers $n\geq0$,…

Let $p(n)$ denote the partition function, and let $\binom{n}{k}$ and $\binom{n}{m}$?

Das et al.'s conjecture. For all $n\geq 0$ and $i\geq 1$, these five congruences hold.

Keith's conjecture. For $n\geq 0$,

Let $M_{C1}(m,n)$ and $M_{C5}(m,n)$ be the coefficients defined by the corresponding two-variable generating functions in the source, with $m\in\mathbb{Z}$ and $n\in\mathbb{N}$, wh…

Let $K(n)$ be defined by its generating function … where $f_k=(q^k;q^k)_\infty$ and $(a;q)_\infty=\prod_{i\geq 0}(1-aq^i)$. Dasappa et al.'s conjecture. For all $n\geq 0$ and…

The non-Rascoe congruence. The source statement does not include the actual congruence, so its mathematical assertion cannot be reconstructed from the supplied span.

The recurrence congruence conjecture.

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…

For positive integers $\ell$ and $k$, let $T_{3^m}(n)$ denote the number of $3^m$-regular partition triples of $n$, where a partition is $3^m$-regular if none of its parts is divis…

Let $P(N)$ be defined by $P(N)=p(n)$ when $N=24n-1$, and $P(N)=0$ if $N<-1$, if $N\not\equiv -1\pmod{24}$, or if $N$ is nonintegral. Let $\left(\frac{a}{b}\right)$ denote the Jacob…

Andrews–Paule's refinement conjecture.

Saikia and Sarma's conjecture. For $n,\alpha\geq 0$,