34 problems

Global chain decomposition conjecture. There exist collections of partitions $\mathcal{C}_{\mu}$, indexed by deficit partitions $\mu$, and a size-preserving involution…

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 $\lambda$ 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…

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…

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

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

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.

Neguț's conjecture. The polynomial $f(a_1,\ldots,a_n)$ has nonnegative coefficients.

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…

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

Conjectured chain decomposition. There exists a sequence of Dyck partitions $\{\gamma_{\lambda,i}:i\in\mathbb{Z}_{>0}\}$ with strictly increasing sizes satisfying all five stated c…

Conjectured decomposition. There exist possibly identical subsets $\mathcal{DV}_{n,\mu}$ and $\overline{\mathcal{DV}}_{n,\mu}$ of $\mathcal{DV}_{n,k}$ such that: (a) each of the tw…

For $a,b\geq 0$, write $b=b'k$ and $a=a'k$ for integers $a',b',k\geq 0$ with $\gcd(a',b')=1$, and let $S_{a,b}(q,t)$ be the q,t-rectangle number. Let $[j]_{q^k}$ denote the q-integ…

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…

For $a,b\geq 0$, define the q,t-rectangle numbers … Here $\mathsf{area}^*_{b,-a}$ is the adjusted area statistic defined using the minimum of the $(b,-a)$-levels. Joint symmetry co…

Joint symmetry conjecture. For all $r,s\in\mathbb{Z}$ and all $a,b\geq 0$,

Let $\mathsf{Cat}_{a,b}(q,t)$ be the rational $q,t$-Catalan number and let $[m]_q$ and $\genfrac{[}{]}{0pt}{}{a+b}{a,b}_q$ denote the $q$-integer and $q$-binomial coefficient. Rati…

Rational $q,t$-Catalan symmetry conjecture.

The combinatorial Hilbert-series symmetry conjecture. The series is symmetric in $q$ and $t$ for all $m$ and $n$:

Let $Y_{m/n}$ be the set of $m/n$ Dyck paths, let $v(D)$ be the set of internal vertices of $D$, and let $\beta(P)$ be the stated vertex statistic. Let…

Let $Y_{m/n}$ be the set of $m/n$ Dyck paths, let $h_{+}(D)$ be the stated statistic, and define … Let $\widetilde{P}_{n,m}$ be the transformed DAHA element, let $h_n$ be the compl…

Let $m$ and $n$ be coprime positive integers, let $\mathcal{P}_{n,m}(u,q,t)$ denote the superpolynomial, and let $D$ range over lattice paths below the main diagonal in an…

Let $\Phi$ be a root system, let $\mathcal{M}(\Phi)$ be its diagonal coinvariant ring, and define its bigraded Hilbert series by … Let $\mathcal{I}(\Phi^+)$ denote the order ideals…