415 problems

Prasad and Ram's conjecture. For any $T\in M_{3m}(\mathbb{F}_q)$,

For $\alpha\in Q^{\geq 0}$, let $\mathfrak{J}_\alpha$ be the Toda eigenfunction and let $(q)_\alpha$ be the associated $q$-factorial. A polynomial is unimodal if its coefficient se…

Let $\pow\geq2$, $k\geq1$, and let $a_1,\ldots,a_k,b$ be positive integers. Write $[r]_q:=1+q+\dots+q^{r-1}$ and, for an integer $\pow$, write…

Let $\Gamma$ be the set of three-colored partitions, and let $A^\ast_1(m,n)$ count the partitions of $n$ in $\Gamma$ having no occurrences of $1_{\textcolor{Red}{\bf R}}$, where…

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…

Generalized Andrews identity.

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 $c(n)$ 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…

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…

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

Determinant conjecture.

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…

Hirschhorn–Sellers conjecture. For all $n\geq 1$ with $3\nmid n$,

Radial limit conjecture. For each $b\in\operatorname{Spin}^c(M)/\{\pm1\}$, there exist topological invariants $\Delta_b\in\mathbb{Q}$, $c\in\mathbb{Z}_{\ge0}$, and…

For any nonnegative integer $n$, define … Let $Q_2(n)$ denote the partition function in the source. Weaker $Q_2$ truncated-sum inequality. For $n\geqslant0$ and $k>0$, … The source…

For any integer $n$, define … Let $b_6(n)$ denote the $6$-regular partition function, and let $(\rho_j)_{j\geqslant0}$ be the coefficient sequence in the source. 6-regular partitio…

For any integer $n$, define … Let $b_6(n)$ denote the $6$-regular partition function. 6-regular partition inequality. For $n\geqslant0$ and $k>0$, … with strict inequality if…

For $n,k\geqslant0$, let $p(n)$ denote Euler's partition function, let $Q_2(n)$ denote the partition function used in the source, and let $(\rho_j)_{j\geqslant0}$ be the coefficien…

Let $k_0,\ldots,k_n$ be non-negative integers, let $E_1$ be the set of parts, and let $\mathcal{Q}^{k_0,\ldots,k_n}_{\mathds{1}}$ be the partitions into parts in $E_1$ whose freque…

Regularized $+p$-surgery formula. When the $+p$-surgery formula of Gukov and Manolescu converges, it should be used; when it does not converge, the displayed identity should regula…

Higher-order modified Alder conjecture. For all $d,n\geq 1$,

Kang and Park's conjecture. For all $d,n\geq 1$,

Let $p$ be a prime with $p\geq 5$. Define the $q$-analogue of the trinomial coefficient by … where $[n]_q=(1-q^n)/(1-q)$ and $\lfloor x\rfloor$ denotes the integral part of the rea…

Catalan triangle Laurent polynomial and positivity conjecture. This expression is a Laurent polynomial in $q$, and it has non-negative integer coefficients when…

Let $R_{n,k}(q,t)$ be the $(q,t)$-Schröder polynomial obtained by weighting little Schröder paths with the statistics defined above, and let $[m]_q$ and $[m]!_q$ denote the $q$-int…