44 problems

For positive integers $d$, define $h_d^{(a)}(n)$ as the number of partitions of $n$ into parts that are $d$-distinct and at least $a$, and define $\ell_m^{(m_1,m_2)}(n)$ as the num…

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…

Generalized Andrews identity.

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…

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…

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:

Andrews's conjecture. For every positive integer $n$,

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

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.

Bressoud's conjecture.

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…

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 $k\geq 3$, 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…

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…

Let $\lambda$ range over partitions, let $h^{\lambda}(u)$ denote the hook length of a cell $u\in\lambda$, and let $c_{sp}^{\lambda}(u)$ be its symplectic content. For an indetermin…

Let $S_m(\rho\mid\sigma)$ denote the finite-level $(z,q)$ sums, and let $R_1$--$R_4$ be the relations listed for the relevant moduli. The R1--R4 sufficiency conjecture. In each mod…