17 problems

A hull configuration is a point configuration lying either on a line segment or on the boundary of a convex polygon; a blank side is an edge of the convex hull containing only two…

Bessis's modified-action CSP conjecture. The triple

Let $W$ be a finite Coxeter group, let $c$ be a Coxeter element, and let a $c$-cluster $C$ have upper roots among its roots. Associate to $C$ the intersection of the hyperplanes or…

Let $W$ be a finite Coxeter group of rank $n$, let $c$ be a Coxeter element, and let $(\alpha_1,\ldots,\alpha_k,\beta_1,\ldots,\beta_{n-k})$ be a $c$-cluster, where the $\alpha_i$…

F–M triangle transformation conjecture. The two generating functions are related by the invertible transformation

Let $W$ be a finite, crystallographic Coxeter group with Coxeter number $h$, let $m$ be the parameter from the preceding theorem, and let ${\sf P}_+$ be the positive Panyushev map…

Combinatorial $H$-triangle conjecture. For positive integers $m,n,t$,

Enumeration conjecture. The number of geometric multi-chains of $t$-filters $(V_m,V_{m-1},\ldots,V_1)$ is

For $m\geq 1$, let $C=(x_1,x_2,\ldots,x_m)$ be an $m$-multichain in $\mathcal{O\!N\C}_{2n+1}$, with extended delta sequence $\delta_o(C)=(d_0;d_1,d_2,\ldots,d_m)$ defined by…

Let $n$ be a positive integer. A meandric partner of a partition $\pi\in\mathrm{NC}(n)$ is a partition $\rho\in\mathrm{NC}(n)$ such that the associated meandric system…

Let $k$ and $n$ be positive integers, and let $\lambda=\lambda_1\lambda_2\ldots\lambda_\ell$ be a type with $\lvert\lambda\rvert$ denoting its size. A small $(k,k)$-Fuss-Schröder p…

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…

Let $\Phi$ be a noncrystallographic root system of rank $n$. Define the antichain generating function … Let $N\!C(\Phi)$ be the noncrossing partition lattice and let … be its Möbiu…

Let $W$ be a real reflection group of rank $n$ with Coxeter number $h$, let $V$ be its reflection representation, and let $k\geq 1$. Let $\theta_1,\dots,\theta_n$ be a homogeneous…

Let $G_T(n)$ be the Hurwitz graph whose vertices are reduced factorizations of the reverse permutation in $S_n$ into transpositions, with edges given by local Hurwitz moves. Diamet…

Fibonacci recurrence-coefficient conjecture. If $\mu_n=NCP_2(n)$, then $b_0=b_1=\lambda_1=1$, and for $n\geq2$,

Let $\Phi$ be a finite root system of rank $n$, let $m$ be a positive integer, and let $F^m_\Phi(x,y)$ be the $F$-triangle of the generalised cluster complex. Let $NC^m(\Phi)$ be t…