52 problems

Let $\lambda \vdash n$ be a partition. Write $\operatorname{Ev}(\lambda)$ for the set of partitions obtained from $\lambda$ in the construction used in the paper, let…

Let $\mathcal C_k$ denote the limit shape of the random-core growth process for fixed $k$. Let $D_k$ be the piecewise-linear curve with vertices … where $\gamma$ is the scaling con…

Symmetry of complements. For all $\lambda\in\mathcal R_k$, $\pi(\lambda)=\pi(\lambda^{\mathrm{comp}})$. The conjecture is supported in the paper by computations for $k\leq 6$.

Let $\lambda$ and $\mu$ be partitions, and let $\lambda\cup\mu=(\nu_1,\nu_2,\nu_3,\ldots)$ be the partition obtained by rearranging all parts of $\lambda$ and $\mu$ in weakly decre…

Let $\lambda$ be a partition and let $rgeq 0$. For the Pieri coefficients $c_i^{\alpha}(\lambda)$, define … Let $d_\rho(\lambda)$ and $z_\rho$ denote the quantities associated with a…

Let $\Lambda=(\lambda_1,\dots,\lambda_d)$ be a finite sequence with sum … Define the polynomial $F_\Lambda(x)$ by $F_\emptyset(x)=1$ and, for $1\leq d=\sharp(\Lambda)<\infty$, … Fo…

Let $I$ be the index set of the affine type ${\tt A}_1^{(1)}$, let $\theta\in\mathbb{Z}_{\geq0}I$, and let $\mathcal{NP}_{\mathrm{2res}}(\theta)$ be the set of skew diagrams…

Let $\mathcal{P}$ be the set of partitions, let $\mathcal{P}_{\mathrm{2reg}}$ be the set of 2-regular partitions, and let…

Let $\overline{\zeta}_n$ be the symmetrized Mordell–Tornheim zeta value of order $n$, and let $p(n)$ denote the partition function. A monomial has total weight $n$ when the sum of…

Let $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ be a partition, and define … For a partition $\lambda$, let $z_\lambda$ be the standard symmetric-function multiplicity factor;…

Let $\mathcal P(n)$ be the set of partitions of $n$, let $\mathbf t\in\mathcal P(n)^3$, and let $g(\mathbf t)$ denote the corresponding Kronecker coefficient. Define … Let $F_n$ be…

For a partition $\lambda=(\lambda_1,\ldots,\lambda_\ell)$, let $\operatorname{pre}_2(\lambda)$ be the partition whose parts are the products $\lambda_i\lambda_k$ for…

For a partition $\lambda=(\lambda_1,\ldots,\lambda_\ell)$, let $\operatorname{pre}_i(\lambda)$ be the partition whose parts are all products $\lambda_{i_1}\cdots\lambda_{i_i}$ with…

Let $n$ be a positive integer, let $\frac{r}{s}$ be a term of the $n$-th Farey sequence, and let $\tilde{M}_{r/s}$ and $\tilde{M}^{\prime}_{r/s}$ be the compositions of generalized…

Partition-number conjecture. The sequence

Let $n$ be a positive integer, and let a partition of size $n^2$ be a partition whose diagram contains $n^2$ cells. For such a partition, consider the game trees of Downright and L…

Let $\rho_k$ denote the staircase partition and let $\delta_\ell$ denote the square partition. Their sizes are $|\rho_k|=\binom{k}{2}$ and $|\delta_\ell|=\ell^2$. For a self-conjug…

Fully symmetric Kronecker asymptotic conjecture.

Model-independent q-bracket conjecture. One has

Low-dimensional character conjecture. If

Longest minimal length partition conjecture. Given $c\in(0,1)$, the set $\Omega$ maximizing $L(\Omega,c)$ under the constraint $|\Omega|=v_d$ is the ball. Given $n>1$ and…

Let $\mathscr{P}_r$ be the set of $r$-dimensional partitions, including the empty partition. For $\lambda\in\mathscr{P}_r$, let $A_{\lambda}$ be the coordinate ring of the Haiman n…

Same-singularity-type conjecture. Then the germs

q,t-symmetry conjecture. The polynomial $C_{\lambda}(q,t)$ is $q,t$-symmetric.

Spectral formula conjecture. The eigenvalues of $J^{[\nu]}$ are