45 problems

Let $s$ be an integer with $s\geq 1$, and consider $(s,s+1,s+2)$-core partitions. The Amdeberhan conjecture. If $s=2t-1$, the largest size is … if $s=2t-2$, the largest size is … T…

Let $s_1$ and $s_2$ be coprime positive integers, and consider the finite set of $(s_1,s_2)$-core partitions. The Armstrong conjecture. The average size of these partitions equals…

Let $n$ be a positive integer, and let a partition be self-conjugate if it equals its conjugate. Its Durfee size is the side length of its largest square contained in its Young dia…

Let … . Negut positivity conjecture. The coefficients in the Negut formula are positive whenever the partition lies under a certain convex curve. This is presented as Conjecture 7.…

Let $s$ be a positive integer, and consider $(s,s+2)$-core partitions with distinct parts. The Straub conjecture. The largest size of such a partition equals … The conjecture was f…

Let $\lambda$ be a strict partition, and let $Q_{\lambda}$ be the Schur Q-function. Give each power sum symmetric function degree $1$, so that $\deg(p_{\nu})=\ell(\nu)$. Let…

Let $a_n$ be the scalar recurrence described in the paper, and let … be its layered expansion, where the $A_k^{(n)}$ are the distinct integer layers. Let the subset of partitions o…

Maximizer conjecture. At most one part of $\lambda$ is the sum of a part in $\mu$ and a part in $\nu$, while the other parts of $\lambda$ are distributed in $\mu$ and $\nu$.

Rank-factorization conjecture. For any partition $\lambda$ of rank $r$, there is a polynomial $Q_\lambda(z;q,t)\in\mathbb{Z}_{\geq 0}[z,q,t]$ such that

Let $k$ be an integer with $k>1$, and let $\lambda$ be a partition. For the Ferrers board $B_\lambda$, let $R_k(\lambda;q)$ be its Garsia–Remmel $q$-rook number, and call this poly…

Garsia–Remmel's conjecture. For every partition $\lambda$ and every admissible rook number $k$, the polynomial $R_k(\lambda;q)$ has a unimodal sequence of coefficients. This conjec…

Let $\lambda$ be a partition and let $\mu$ be a partition of the same number of boxes. A Latin tableau of shape $\lambda$ and type $\mu$ is a filling of the Young diagram of…

Let $\lambda<_D\mu$ be randomly chosen partitions, where $<_D$ is the dominance order. Near-half probability conjecture. The probability that $\lambda\leqslant_I\mu$ in the immersi…

Let $n$ be a positive integer and let $[(1^n),\mu]$ be an interval in the reverse lexicographic order on partitions of $n$. Let $p_{[(1^n),\mu]}$ denote the symmetric-function expr…

For a partition $\mu$ of $n$, let … be its lower interval in the immersion poset, and let $\lessdot_I$ denote a cover relation. The $(n-2,2)$ interval conjecture states that, for…

Let $\mathcal{E}_\Lambda^\bullet$ be the category whose objects are indexed by nonnegative integers, with morphisms represented by the elements $\rho_\lambda$ associated to partiti…

Let a triangular partition be the maximal partition fitting below a line joining $(r,0)$ and $(0,s)$ for non-negative real numbers $r$ and $s$, and let the associated $q,t$-enumera…

For $m\ge 7$, let $(m^m)$ be the square partition of $m^2$, let $\mu'$ denote the conjugate partition of $\mu$, and define … Write $g(\lambda,\lambda,\mu)$ for the Kronecker coeffi…

Let $g$ be a non-negative integer and consider the $g$-loop quiver. Rodriguez-Villegas introduced a refinement of its Kac polynomials indexed by partitions; denote these refined Ka…

Let $\lambda,\mu,\nu\vdash n$ be partitions, and let $(\lambda+1)\cup1$ denote the partition obtained by adding $1$ to the first part of $\lambda$ and adjoining a part $1$. Let…

Opposite-property conjecture. The chains $\mathcal{C}_{\mu}$ satisfying these conditions can be chosen to satisfy the opposite property.

Balanced-partition product conjecture. If $\lambda$ is balanced, then

Converse CDE conjecture for shifted diagrams. The interval $[\emptyset,\lambda]$ has the CDE property if and only if $\lambda$ is balanced or trapezoidal.

Converse CDE conjecture. The interval $[\emptyset,\lambda]$ has the CDE property if and only if $\lambda$ is balanced.

The local chain conjecture. There is a set $mathcal{L}$ of local chains of deficit $k$ and an involution mathcal{S}to mathcal{S}^ on $mathcal{L}$ such that: every…