19 problems

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 $c(n)$ denote the coefficient sequence under consideration, and let $k,n\in\mathbb{N}_0$. The infinite-family congruence conjecture for $c(n)$. … … … The authors propose these…

Mod 4 congruence conjecture. For every $n\geq0$ and every integer $j$ with $0\leq j\leq\ell-2$,

Let $a(m)$ be defined by the paper's generating function, and suppose that $m$ is a positive integer such that … Write the prime factorization of $4m+3$ as … where each…

Let $N(r,2;n)$ denote the number of partitions of $n$ whose rank is congruent to $r$ modulo $2$, for $r\in\{0,1\}$. Hou–Jagadeesan's conjecture. If $r=0$ (respectively, $r=1$), the…

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…

The finite sign law. For every $n\geq 0$,

Let … where $(a;q)_n=\prod_{j=0}^{n-1}(1-aq^j)$. Andrews' unboundedness conjecture. … The coefficients $S(n)$ arise from the mock theta function studied by Andrews, Dyson, and Hick…

Let $m,m'\in\mathbf N$ and $j,j'\in\mathbf Z$. For $m\in\mathbf N$ and $j\in\mathbf Z$, let $\overset{N=3}{CH}{}^{(+)[K(m),j]}$ denote the $\mathbf C((q^{\frac12}))$-linear span of…

Let $N_A(n)$ be defined by … Let $p\not\equiv7\pmod8$ be an odd prime, let $\delta_p$ satisfy … and let $k,n\in\mathbb Z$ with $\left(\frac{k}{p}\right)=1$. Kim–Lim–Lovejoy's…

Let $\operatorname{spt}(n)$ denote the number of occurrences of the smallest part among the partitions of $n$. Let $\ell>3$ be prime with $\ell\not\equiv23\pmod{24}$, and define ……

Let $v(n)$ count odd-balanced unimodal sequences of size $2n+2$, and let $v(m,n)$ count those with rank $m$. Let $\ell\not\equiv-1\pmod8$ be prime and let $k$ be a positive integer…

Bryson–Ono–Pitman–Rhoades conjecture. For all $n$,

Let $p_{\xi}(n)$ denote the coefficients of the Gordon–McIntosh mock theta function $\xi(q)$, so that $\xi(q)=\sum_{n\geq 0}p_{\xi}(n)q^n$. Generating-function congruence. One has…

Let $g(q)$ be a mock theta function with integer coefficients, and let $c(g;n)$ denote the coefficient of $q^n$ in its series expansion. Say that $g$ is of parity type…

Let $L(\Lambda)$ be an integrable nonzero-level module over the affine Lie superalgebra in the surrounding section, with $\Lambda\in\Omega_k$ or $\Lambda\in\Omega'_k$. Define…

Radial-limit identity. The following identity should hold:

For fixed integers $a,b,A,B$, let $f_{a,b,A,B,h,k}$ be the modular forms used to cancel the singularities of $g_2(\zeta_b^a q^A;q^B)$ as $h/k$ ranges over $\mathbb{Q}$. Rhoades' fi…

Let $f$ be a mock theta function in Ramanujan's sense, and let $\xi$ be a root of unity. Radial limits are taken along the line through $\xi$, either inside or outside the unit cir…