56 problems

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…

Bressoud's conjecture.

Let $q$ be a formal variable, and for integers $a_1,\dots,a_r$ define … For $i,j\geq 0$, let $(q;q)_i$ and $(q^3;q^3)_j$ denote the corresponding finite $q$-Pochhammer symbols. And…

Let $\lambda=(\lambda_1,\ldots,\lambda_r)$ be a partition. Bessenrodt–Morris–Olsson's conjecture. For $p=5$, the labels for irreducible modular spin representations are the partiti…

Let $k\geq1$, $1\leq\ell\leq k$, $1\leq i\leq k+1$, $1\leq i'\leq\ell+1$, and $|q|<1$. Let $C_k$ be the Cartan matrix of type $A_k$, let $B_{\ell-1}$ be the inverse Cartan matrix o…

Genus-one product identity.

Genus-one partition identity. The following combinatorial identity holds:

Rogers–Ramanujan type character identity. For $n\geq 1$ and $0\leq b\leq 2n$, the identity

Generalized Andrews identity.

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 $FO_{j,k}(n)$ be the number of partitions of $n$ with exactly $j$ parts divisible by $k$, and let $FD_{j,k}(n)$ be the number of partitions of $n$ with exactly $j$ parts appear…

Cylindric-partition hierarchy conjecture.

Finite companion Bressoud conjecture.

Finite companion Andrews–Gordon conjecture.

Lepowsky–Zhu's Empirical Hypothesis. For $j\geq 0$ and $1\leq i\leq k$, one has

Fix an integer $d\geq 1$. For an unordered partition $(d_1,\ldots,d_r)\vdash d$ into strictly positive parts, let $\#\operatorname{Aut}(d_1,\ldots,d_r)$ be the order of its automor…

For $|q|<1$, let $(a;q)_n$ denote the $q$-Pochhammer symbol, defined by $(a;q)_n=\prod_{k=0}^{n-1}(1-aq^k)$ and $(a;q)_\infty=\prod_{k=0}^{\infty}(1-aq^k)$; for multiple parameters…

Let $p,q\in\mathbb{N}$ with $(p,q)=1$, and let … be endowed with the action of the bosonic vertex operator $V_{\sqrt{p/q}}(z)$. A Feigin–Stoyanovsky basis conjecture. The space…

Let $\ell\geq 2$ and $k_0,k_1,\dots,k_\ell\in\mathbb N_0$, with $k=k_0+\dots+k_\ell>0$. An $(k_0,k_1,\dots,k_{\ell-1},k_\ell)^{\operatorname{odd}}$-admissible colored partition is…

Fix $m=3k+\{-1,0,1\}$ with $k\geq3$. Let $\mathcal S_m$ be the set of relations obtained from the coupled $q$-difference equations in terms of the functions $S_m(\rho\mid\sigma)$.…

Fix $m=3k+\{-1,0,1\}$ with $k\geq3$. Let $\mathcal S_m$ be the set of relations obtained by writing the coupled $q$-difference equations for the profiles $c$ with $|c|+3=m$ in term…

Let $m\geq5$ be a modulus of the form $m=3k+\{-1,0,1\}$ with $k\geq3$, and let $S_m(\rho\mid\sigma)$ be the multiple-sum functions indexed by integer vectors…

Let $F$ be the set of the following 13 partitions: … Define … … For $1\leq a\leq3$, let $\mathsf L_a$ be the set of partitions that do not begin with any $\boldsymbol c\in I_a$ and…

Let $q$ be a formal variable, let $\omega$ be a primitive third root of unity, and define … … Here $(a;Q)_\infty=\prod_{j\geq0}(1-aQ^j)$, multiple parameters are interpreted multip…

Let $\lambda$ range over partitions, let $h^{\lambda}(u)$ denote the hook length of a cell $u\in\lambda$, and let $c_{sp}^{\lambda}(u)$ and $c_{O}^{\lambda}(u)$ denote its symplect…