54 problems

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…

Haiman's q,t-Catalan evaluation conjecture. The specialization at $q=t=1$ equals the classical Catalan number:

The compositional shuffle conjecture. For any composition $\alpha$,

For coprime positive integers $a,b$, let $C_{b,-a,a,b}(q,t)$ be the generalized q,t-Catalan polynomial defined using the sweep map, and let $[j]_q$ denote the q-integer and…

For $a,b\geq 0$, let $S_{a,b}(q,t)$ be the q,t-rectangle numbers defined from words with $a$ letters $\mathrm{N}$ and $b$ letters $\mathrm{E}$, and let $p_n$ and $s_{(1^n)}$ denote…

Let $C_n(q,t)$ denote the $q,t$-Catalan polynomial and set … For integers $n\geq 1$ and $d$ with $0\leq d\leq \lfloor M/3\rfloor$, consider the coefficients of monomials on the dia…

Qu and Zhang's area-depth symmetry conjecture. This variant of $\widetilde{C}_M(q,t)$ remains symmetric in $q$ and $t$.

Let $\vec{k}$ be a vector of positive integers and define the area-bounce polynomial … A polynomial is $q,t$-symmetric when $F(q,t)=F(t,q)$. The $(a,b,\ldots,b)$-symmetry conjectur…

Let $\vec{k}$ be a vector of positive integers, let $\mathcal{K}$ be the associated parameter, and define the area-bounce polynomial … Write $\lambda(\mathcal{K})$ for the partitio…

Let $\vec{k}$ be a vector of positive integers, let $\mathcal{K}$ denote the corresponding parameter indexing the relevant $\vec{k}$-Dyck paths, and let…

Let $C_n^{(k)}(q,t)=\left<\nabla^{k}e_n,e_n\right>$ denote the higher $q,t$-Catalan polynomial, and let $k$-Dyck paths be the associated generalized Dyck paths, with area and bounc…

For a vector $\vec{k}$, let $\mu(\vec{k})$ be the partition obtained by rearranging its entries in decreasing order, and define … For positive integers $a$ and $n$, and an integer…

Let $\lambda$ be a triangular partition, and let its top-down tableau be the Young tableau associated with a maximal chain in the $\nu$-Tamari lattice. Say that this tableau is sim…

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…

Assume $d_1\leq\cdots\leq d_n$, set $d_{ij}=\min(d_i,d_j)$, and define … Let $\mathfrak m=(t_1,\ldots,t_n,x_1,\ldots,x_n)$. Generalized Haiman ideal Hilbert-series conjecture. The…

Let $C_n(q,t)$ be the rational function introduced as the $q,t$-Catalan number, let $D_n[X;q,t]$ denote the diagonal coinvariant alternant, and let $e_n$ be the $n$-th elementary s…

Hawkes's degreewise conjecture. For every $d$,

Hawkes's conjecture. The rational $q,t$-Catalan polynomial satisfies

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

For a vector $\mathbf b=(b_1,\ldots,b_l)$, define the generalized $q,t$-Catalan number … Here $|\mathbf b|$ denotes the sum of the entries of $\mathbf b$. Negut's positivity conjec…

Convex-path positivity conjecture. When the vector $\mathbf b$ arises from the highest lattice path below a convex curve as above, the symmetric function…

Neguț's conjecture. If $a_2\geqslant a_3\geqslant \cdots\geqslant a_n\geqslant 0$, then $F(a_2,\ldots,a_n)$ has nonnegative coefficients.

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…

Let $\mathcal{C}_n(q,t)$ and $\mathcal{C}'_n(q,t)$ be the two $q,t$-Catalan polynomials introduced in the source. Divisibility conjecture. We have … for some…

Strong structure conjecture. There exist possibly identical collections $\mathcal{C}_\mu$ and $\overline{\mathcal{C}}_\mu$ of Dyck partitions satisfying conditions (1)--(9): the tw…