638 problems

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

Let $p$ be an odd prime, and for $n\geq 0$ let $(a)_n=a(a+1)\cdots(a+n-1)$ denote the Pochhammer symbol. Van Hamme's (B.2) supercongruence. … This is the stated $p$-adic analogue o…

Let $c(n)$ denote the coefficient sequence under consideration, and let $n\in\mathbb{N}_0$. The congruence conjecture for $c(32n+23)$. … This conjecture is proposed as an open ques…

Let $a_0,\dots,a_n$ be non-negative integers, and let $\operatorname*{CT}_{x}$ denote the constant term in the variables $x_0,\dots,x_n$. Dyson's conjecture. … Dyson's conjecture i…

Merca's conjecture. This theta series has non-negative coefficients. The conjecture refines the established positivity theorem for averaged truncations of Jacobi's triple product i…

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 integers $d\geq 1$, consider the partial fraction decomposition … Here $C_{h,k,\ell}(d)$ denotes the coefficient associated with the reduced root of unity $e^{2\pi i h/k}$ and…

Andrews' local-minimum conjecture. The three numbers

Gukov–Manolescu conjecture. The invariant $F_K$ is obtained by reorganizing this perturbative expansion into an integral two-variable $q$-series.

Positive-definiteness conjecture. Despite the negative signs in the definition of $Q$, the quadratic form $Q$ is positive definite on the cone $C_{\mathbb{R}}$.

Beukers' conjecture.

For nonnegative integers $A,B$, define the $q$-binomial coefficient by … Suppose that $M,N\in\mathbb Z^+$, that $\alpha$ and $\beta$ are positive rational numbers, and that $K$ is…

Andrews–Bachraoui conjecture. There holds

Let $t_o(n)$ be defined by … where $q$ is a complex number with $|q|<1$ and $(a;q)_n$ denotes the standard $q$-Pochhammer symbol. Andrews–El Bachraoui's conjecture. If $8n+9$ has a…

Richmond–Szekeres conjecture. The coefficients satisfy

Let $k\geq 3$, let $c=(c_1,c_2,c_3)$ be a cylindric partition profile with $|c|+3=m=3k+\{-1,0,1\}$, and, using cyclic symmetries, assume $c_1\geq c_2,c_3$. Let…

Andrews' same-sign triple conjecture. There is an infinite sequence with these indicated values and growth condition such that $V_1(N_k),V_1(N_k+1),V_1(N_k+2)$ all have the same si…

Let $P(n,m)$ denote the number of partitions of $n$ with $m$ missing integers, and define … Let $q(n)$ denote the number of partitions of $n$ into distinct parts. Distinct-part bou…

Let … Here $q$ is a complex number with $|q|<1$, so the double Lambert series defines an analytic function in its domain of convergence. Andrews–Dixit–Schultz–Yee conjecture. The f…

Let $a_t(n)$ be the total number of $t$-hooks in all partitions of $n$ into odd parts, and let $b_t(n)$ be the total number of $t$-hooks in all partitions of $n$ into distinct part…

For $k,n\in\mathbb{N}_0$, define $G_{k,n}(q)$ by … Here $\begin{bmatrix}a\\b\end{bmatrix}_q$ denotes the $q$-binomial coefficient, and $A(q)\succeq0$ means that all coefficients of…

The $(A_1,D_{2n+1})$ Macdonald-index conjecture. The Macdonald index is

Bressoud's conjecture.

Let $G(q)$ be the $q$-series … and let $e(n)$ and $o(n)$ denote respectively the numbers of partitions of $n$ into odd parts whose index is even and odd. Coll–Mayers–Mayers conject…

Keith–Zanello's self-similarity conjecture. For a positive proportion of primes $p$, one has