424 problems

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 $DSOME(n)$ 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 $n\geq0$,…

Determinant conjecture.

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 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…

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 $c(n)$ 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 $D'(k,m,n)$ be defined by … where $k$ and $m$ are positive integers. A series is eventually positive when all coefficients are nonnegative from some index onward. The eventual…

Let $D'(k,m,n)$ be defined by … where $k$ and $m$ are positive integers. A coefficient is negative when it is less than zero. The exceptional negative coefficients conjecture for…

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

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…

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…

For every integer $n\ge 1$, define $P_n(q)=\prod_{j=1}^{n}(1-q^{3j-2})(1-q^{3j-1})$ and $c_{n,m}=[q^{3m+2}]P_n(q)^4$ for $0\le m\le 4n^2-1$. After all zero coefficients are omitted…

For every word $w$ in the class defining Hirose's iterated $q$-integrals on the four-punctured projective line, with the prescribed position-dependent $q$-shifts of the parameters,…

For each positive integer $n$, let $\mathcal{O}(n)$ be the set of overpartitions of $n$, and define…

Let $F_1(q),F_2(q),F_3(q),F_4(q)$ denote the four weighted truncated pentagonal-number series appearing in Merca's conjectures. The conjectures assert that $[q^n]F_i(q)\ge 0$ for e…

Prove the eighteen explicit conjectural sum–product identities arising in the modulo-nine Kanade–Russell program: ten identities obtained from reflections of finite forms under…

Let $Z(3)$ denote the finite multiple-zeta value corresponding to $\zeta(3)$ under the Kaneko–Zagier correspondence, defined from Bernoulli numbers, and let $Q$ denote the quotient…

Let $\mathcal{C}_{a,b,m}(q)$ denote the infinite copartition-product generating function of Burson and Eichhorn, and let $\mathcal{C}_{a,b,m;N,M}(q)$ denote its finite version. The…

Let $c\phi_{18}(n)$ denote the number of $18$-colored generalized Frobenius partitions of $n$. Prove that, for every integer $n\geq 0$, $c\phi_{18}(30n+19)\equiv 0\pmod{16}$ and…

For each $i\in\{1,2,3,4,5\}$, prove the corresponding modulo-$9$ Kanade–Russell Rogers–Ramanujan-type sum–product identity, namely…

Andrews–El Bachraoui positivity conjecture. For every positive integer $k$, the series $F_{k,1}(q)$ is positive.