31 problems

Let $\mathrm{DH}_n$ denote the space of diagonal harmonics, viewed as a bigraded $S_n$-module, and let $\operatorname{Frob}(\mathrm{DH}_n)$ denote its bigraded Frobenius characteri…

Dyck-path dimension conjecture. The dimension of $DHA_n$ is given by the number of Dyck paths in ${\cal L}_n$.

Let $\mathcal{D}_{n}^{(3)}$ be the space of polynomials in $x_1,\ldots,x_n,y_1,\ldots,y_n,z_1,\ldots,z_n$ annihilated by every polarized power-sum operator … where…

Let $C_n(q,t)$ be the $q,t$-analog of the Catalan numbers, and let $\operatorname{area}(\pi)$ and $\operatorname{bounce}(\pi)$ denote Haglund's area and bounce statistics on Dyck p…

Let $\mathrm{DH}_n$ be the space of diagonal harmonics, and let $\mathcal F(\mathrm{DH}_n)$ denote its Frobenius image. Let $e_n$ be the $n$th elementary symmetric function, … and…

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…

Polygraph quotient conjecture. The homomorphism $\psi$ is an isomorphism.

Let $\mathrm{DH}_n$ be the space of diagonal harmonics. The diagonal-harmonics dimension conjecture. … The conjecture is one of the numerical consequences of the diagonal-harmonics…

For certain triangular partitions, computations comparing Tamari-interval enumerations with proposed symmetric functions indicate that the expressions for $A_{\lambda}$ may omit Sc…

Let $e_r$ denote the elementary symmetric function, let $e_{n-k}^{\perp}$ be the operator adjoint to multiplication by $e_{n-k}$ under the Hall inner product, and let…

Let $\mathcal{DH}_{n}$ be the space of diagonal harmonics, and let $\nabla$ be the linear operator on symmetric functions whose eigenfunctions are the modified Macdonald polynomial…

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…

Let $\mathcal{E}_{n,n}$ denote the stable character of the $GL_k\times \mathbb{S}_n$-module of multivariate diagonal harmonics, expressed in the Schur-function basis, and let…

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 $\mathrm{DH}_{n,r}$ be the multivariate diagonal harmonics space, with its homogeneous components indexed by multidegrees, and let intervals in the Tamari lattice carry combina…

Diagonal-harmonics inversion conjecture. The specialized Hilbert series is conjectured to satisfy

Pak's conjecture. Eventually, the number of Tesler matrices is bounded below by the dimension of $DH_n$:

Garsia–Haiman conjecture. The function $C_n(q,t)$ is a polynomial in $q$ and $t$ with nonnegative coefficients, and there is a statistic $\mathrm{unknown}(w)$ on $n$-Dyck paths suc…

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 $\mathscr{H}_n^{(r)}$ be the space of higher diagonal harmonics, let $\Dyck{r}{n}$ denote the set of $r$-Dyck paths, and let $\mathcal{P\!F}(\beta)$ be the set of parking funct…