12 problems

Coefficient-Length conjecture. For all triangular partitions $\tau$,

Let $P$ be a unit interval order corresponding to an incomparability graph $G(d)$ and to a lex-maximal part listing encoded by ${\bf a}=(a_1,\ldots,a_n)$. Let $d'$ be the Dyck path…

Fix a partition $\lambda\vdash n$, and let $c_\lambda(q,r)$ be the coefficient of $e_\lambda$ in the Frobenius characteristic $\Phi_{n,r}(q)$. Write $\lambda_i=0$ for $i$ larger th…

Let $\mathrm{DH}_{n,r}$ denote the multivariate diagonal harmonics space, and let $\Phi_{n,r}(q)$ be the Frobenius characteristic of its character, with parameter $q_r$. A symmetri…

Let $c$ be a composition, let $\nabla$ be the nabla operator on symmetric functions, let $C_c=C_{c_1}\cdots C_{c_k}$ for $c=[c_1,\ldots,c_k]$, and let $\mathcal{A}_c$ be the set of…

The diagonal-word generating-function conjecture. For $b\leq a-1$ and every permutation $\tau$, one has

Let $\mathcal{A}_c$ denote the set of parking functions with composition $c$, let $\operatorname{dinv}(PF)$ and $\operatorname{ides}(PF)$ be the diagonal inversion number and inver…

Let $PF$ be a parking function, let $\operatorname{area}(PF)$, $\operatorname{dinv}(PF)$, and $\operatorname{ides}(PF)$ denote its area, diagonal inversion number, and inverse desc…

Let $\mathscr{H}_n^{(r)}(\mathbf{w};1,1,1)$ denote the specialization of the graded Frobenius characteristic of the higher diagonal harmonics at $q_1=q_2=1$. For a partition…

Let $W=G(m,n)$, let $\ell\in\mathbb{N}$, and let $q$ be formal. Define $\mathcal{D}W^{(\ell)}_q$ as the space of polynomials annihilated by all operators $D_{q,\mathbf{d}}$ for whi…

Let $D^{mn-1}(n)$ be the dilated simplex associated with bounded chambers of the $m$-extended Shi arrangement, and let ${\sf shi}^m(A)$ and ${\sf ish}^{-1}(A)$ be the corresponding…

Let $\mathcal D_n^{\mathbf a}$ be the common polynomial zero space in $\mathbb C[\mathbf{x},\mathbf{y}]$ of the operators … for all $k,j\in\mathbb N$ with $k+j>0$. Assume that…