24 problems

Let $D_n^{(r)}$ be the set of all $r$-fans of Dyck paths of length $2n$, and define … where $[m]_q=1+q+q^2+\cdots+q^{m-1}$. Let $C_{2n}$ be the cyclic group of order $2n$ acting on…

Bessis's modified-action CSP conjecture. The triple

Let $s$ be a positive integer, let $\Phi$ be a root system of rank $n$ with Coxeter number $h$ and exponents $e_1,\ldots,e_n$, and let $X$ be the set of facets of the generalized c…

Let $W$ be a well-generated finite complex reflection group with degrees $d_1,\dots,d_n$ and Coxeter number $h$, let $d$ be a regular degree, let $W'$ be the centralizer of a $d$-r…

Let $P$ be a finite graded poset of rank $r$ for which the roots of $\aΩ_P(m)$ are all integers or half-integers, and in the latter case possessing an order two automorphism. Let…

Cyclotomic Catalan conjecture.

Stanton's conjecture. The polynomials $f(n,k;q)$ and $a_\mu(q)$ satisfy

Adams et al.'s conjecture. The number of weak excedances that are also mid-points of a decreasing subsequence of length $3$ (Statistic $373$ in FindStat) is equidistributed with th…

Cyclic sieving conjecture. For any positive integer $n$ and $0\leq j\leq n$, the triple

Short-distance inversion conjecture. For $n\geq 2$ even, the number of inversions of distance at most $3$ (Statistic $494$) exhibits the cyclic sieving phenomenon under involutions…

Weak-excedance mid-point conjecture. The number of weak excedances that are also mid-points of a decreasing subsequence of length $3$ (Statistic $373$) is equidistributed with the…

Coxeter-length difference conjecture. The difference in Coxeter length of a permutation and its image under the Simion–Schmidt map (Statistic $123$) exhibits the cyclic sieving phe…

Thomas–Williams' cyclic sieving conjecture. The triple

Staircase flagged-tableaux cyclic sieving conjecture. The triple $(\mathrm{FT}(sc_n,b),\langle\operatorname*{Pro}\rangle,Cat_{\ell}(x))$ exhibits the cyclic sieving phenomenon.

Let $V(n)$ be the poset in the source, let $\mathcal{L}(V(n))$ denote its set of linear extensions, and let $\operatorname{Pro}$ denote promotion. Hopkins–Rubey's conjecture. For e…

Let $\lambda/\mu$ be a skew shape, let $m$ be a positive integer, and let $k$ be a positive integer. Write $m\lambda/m\mu$ for the skew shape obtained by multiplying every part of…

Transpose-and-rowmotion cyclic sieving conjecture. For every $k\in\mathbb Z$,

Minuscule P-partition cyclic sieving conjecture. For every $\ell\geq 1$, the triple

Let $P$ be one of the two minuscule posets associated to an exceptional Lie algebra of type $E$, and let $k\in\mathbb{Z}_{\geq 0}$. Write ${\mathsf{PP}}^k(P)$ for the plane partiti…

Let $W$ be a Weyl group with Coxeter number $h$, let $m=sh+1$, and let $NC^{(s)}(W)$ denote the set of multichains $w_1\leq\cdots\leq w_s$ in the $s$-divisible noncrossing partitio…

The t=-1 phenomenon conjecture. The value $\EuScript M_{\Delta(1)}(-1)$ equals the number of upper ideals $I$ satisfying $I^*=I$.

Cyclic sieving conjecture. If $k$ is equal to $1$, $2$, or a power of an odd prime, then $|Q_i(-1)|$ counts the number of self-complementary dichotomies with automorphism group…

Let $W$ be a finite crystallographic Coxeter group with Coxeter number $h$, let $J(\Phi^+(W))$ be the order ideals of its positive root poset, and let $d_1,d_2,\ldots,d_n$ be the d…

For $n\geq 2$, label the vertices of the centrally symmetric polygon ${\Bbb P}_{2n}$ clockwise by $1,2,\dots,n,\bar{1},\bar{2},\dots,\bar{n}$. Let $V^D(n)$ be the quotient space de…