18 problems

Let $M(m,n)$ and $N(m,n)$ denote respectively the number of partitions of $n$ with crank $m$ and rank $m$. Rank and crank log-concavity conjecture. The inequalities … hold for…

For $n\ge0$, let $\chi(\mathcal S)$ be the number of distinct parts occurring in a set of partitions $\mathcal S$, and let $\tau(\mathcal S)$ count parts with multiplicity. Let…

Let $\mathcal P(n)$ be the set of partitions of $n$, let $\mathcal P[m,n]=\biguplus_{i=m}^n\mathcal P(i)$, and let…

Let $u(m,n)$ denote the number of strongly unimodal positive integer sequences of weight $n$ and rank $m$. Bringmann–Jennings-Shaffer–Mahlburg–Rhoades' conjecture. For…

Let $W_{\rm size}(n)$ be the size-ensemble norm statistic and let $\widehat{W}_{\rm size}(n)$ be the corresponding non-cumulative reciprocal supernorm statistic. Size-ensemble norm…

Let ${\rm Max}(n)$ be the ensemble of partitions with largest part $n$, and let $\widehat{W}_{\rm max}(n)$ denote its non-cumulative reciprocal supernorm statistic. Max-part-ensemb…

Let ${\rm Per}(n)$ be the perimeter ensemble of partitions indexed by $n$, and let $\widehat{W}_{\rm per}(n)$ denote its non-cumulative reciprocal supernorm statistic. Perimeter-en…

Borozenets' conjecture. The following chains of inequalities hold:

Let $NT(r,m,n)$ denote the total number of parts in partitions of $n$ whose rank is congruent to $r$ modulo $m$, and let $M_{\omega}(r,m,n)$ denote the total number of ones in part…

Let $n$ be a positive integer, and let $p(n)$ denote the number of partitions of $n$. A partition $\lambda=(\lambda_1,\ldots,\lambda_j)$ of $n$, written $\lambda\vdash n$, has a fi…

Let $NT(i,5,k)$ denote Beck's partition statistic evaluated at residue class $i$ modulo $5$. For integers $0\leq i<j\leq 4$, consider the proportion of positive integers $k$ up to…

Let $M_{\omega}(i,5,k)$ denote Beck's partition statistic evaluated at residue class $i$ modulo $5$. For integers $0\leq i<j\leq 4$, consider the proportion of positive integers…

Garvan k-rank unimodality conjecture. For each integer $k\ge2$ and integer $n\ge n_u(k)$, the sequence

Let $N(m,n)$ denote the number of partitions of $n$ with rank $m$, and recall that a finite sequence is unimodal if it weakly increases to some term and then weakly decreases. Chan…

Keith's quadratic correction conjecture. The exact term that must be added to the $b^2$ coefficient of $q^n$ in $C'_{1-b,2}(q)$ to obtain the corresponding coefficient in $H_2(q)$…

Keith's coefficient conjecture. For every $j$, the constant and linear terms in $b$ of the coefficients of $q^n$ in the two generating functions agree; equivalently,

Fixed-core generating function conjecture. One has

Let $\overline{N}^+_{k}(n)$ denote the $k$-th positive symmetrized rank moment of overpartitions, for integers $k\ge 1$ and $n\ge 1$. The higher-order overpartition moment inequali…