38 problems

Brady–McCammond conjecture. The metric space

Let $k$-noncrossing partitions be set partitions containing no crossing of size $k$, and let the sequence count such partitions as a function of the underlying set size. Bousquet-M…

Let $k$-noncrossing partitions be set partitions in which no $k$ arcs mutually cross, and let $k\geq 4$. A sequence is P-recursive if it satisfies a linear recurrence with polynomi…

Equivariant-bijection conjecture. Assuming the $m=1$ case of the preceding conjecture, there exists a $\mathbb Z/2h\mathbb Z$-equivariant bijection

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 $\alpha_i$…

Let $\Phi$ be a root system of rank $n$, let $\Delta^m(\Phi)$ be the $m$-generalised cluster complex, and let $NC^m(\Phi)$ be the $m$-divisible non-crossing partition poset. Define…

Simion's refined determinant conjecture. For every positive integer $n$,

Simion's type-B determinant conjecture. For every positive integer $n$,

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

Self-dual coherent omega-equivalence conjecture. The augmented chain complex $C$ is the linearisation of a polygraph presenting a self-dual model of the coherent walking $omega$-eq…

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…

Let $S_n$ be the symmetric group, let $\gamma_{p,q,l}$ be the permutation defining the reference circles, and let $\lesssim$ be the relation such that, for each block $B$ of…

Sign conjecture. The sign of $\lambda_{\mathcal{NCP}}(J_n)$ is

Let $\mathsf{NCBipartEven}_N$ be the noncrossing bipartite-even category, and let $\mathsf{NCPair}_n$ denote the noncrossing-pair category. Product decomposition conjecture. Motiva…

Irreducible-partition ratio conjecture. In the case $k=1$, $\bar m=2$,

Let $\Phi$ be a finite root system with Weyl group $W$, let $c$ be a Coxeter element, let $\mathcal{A}(\Phi^+)$ be the antichains of the positive-root poset, and let…

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

Fix simple reflections $s_0,s_1,\ldots,s_{kn-1}$ in the hyperoctahedral group of type $B_{kn}$, and form the type $B_n$ $k$-indivisible noncrossing partitions from prefixes of fact…

Let $\mathcal{N\!C}_{N;k}$ be the poset of $k$-indivisible noncrossing partitions, ordered by refinement. Its order complex is obtained by removing its least and greatest elements.…

For $m\geq 1$, let $\mathcal{O\!N\C}_{2n+1}^{(m)}$ be the poset of $m$-multichains defined by the extended delta sequence order. Let $\overline{\mathcal{O\!N\C}}_{2n+1}^{(m)}$ be o…

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 $n\geq 2$. Let $\operatorname{RG}_\mathrm{nc}(n)$ denote the set of noncrossing restricted-growth words of length $n$, and let $\operatorname{RG}_\mathrm{nc}(n,k)$ denote those…