14 problems

For $t \geq 4$, let $f_{t;n}(x)$ be the scaled mass function associated with the distribution of $\hat{Y}_t(n)$, and let $g_t(x)$ be the corresponding limiting function. Pointwise…

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 $W_{\rm size}(n)$ 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 $M(m,n)$ 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…

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

Fixed-core generating function conjecture. One has

Let $s$ and $t$ be coprime positive integers. An $(s,t)$-core is a partition that is simultaneously an $s$-core and a $t$-core; its size is the sum of its parts. Armstrong's conjec…

Dyson's crank limiting-shape conjecture. As $n\to\infty$,