10 problems

Let $p_t(n)$ denote the number of $t$-multipartitions of $n$. Let $a$ and $t$ be odd integers, with $3\mid t$ whenever $3\mid a$. Define $b$ by … when $3\mid t$, and by…

Let $\boldsymbol{\kappa}=(\kappa_1,\kappa_2)$ be a bicharge, let $\boldsymbol{\lambda}\in\mathcal{MP}^{\boldsymbol{\kappa}}$, and let $\boldsymbol{\lambda}_{\mathrm K}$ denote its…

Let $\Pi$ be the set of pairs of partitions, let $i\in\mathbb{Z}_2$, and let $\mathcal{X}_i:\Pi\rightarrow\mathcal{UL}$ be the associated map to upper ledge diagrams. If…

Let $\mathbf{s}$ be a multicharge of the form … Let $\lambda$ be a cylindric $l$-partition, and let $m(\lambda)$ denote the unique $l$-partition with minimal $b$-invariant occurrin…

For $t\geq1$, let $p_t(n)$ be the $t$-multipartition function, and let $\delta_t$ denote the density of those $n$ for which $p_t(n)$ is odd, when this limit exists. Density-transfe…

Judge–Keith–Zanello conjecture. The density $\delta_t$ exists and equals $1/2$ for every positive odd integer $t$. Equivalently, if $t=2^k t_0$ with $t_0\geq1$ odd, then $\delta_t$…

Let $s$ be a positive integer and let $a,b$ satisfy $0\leq a,b<s$. For parameters $r$ and $c$, write $\mathcal{C}_{(r\,|\,0,c)}$ for the corresponding set of bipartitions, and let…

Let ${\boldsymbol\lambda},{\boldsymbol\mu}\in\mathscr{P}_{n}$. For the polynomials $d_{{\boldsymbol\lambda}{\boldsymbol\mu}}(q)$ and the defect…

Multipartition equidistribution conjecture. The density $\delta_t$ exists and equals $1/2$ for every odd positive integer $t$. Equivalently, if $t=2^k t_0$ with $t_0\geq 1$ odd, th…

Let $\mathcal{P}_r(n)$ be the set of $r$-partitions of $n$, let $\mathcal{K}_r(n)$ be the set of Kleshchev $r$-partitions with respect to $(e;\mathbf{Q})$, and let an $r$-partition…