27 problems

Let ${\mathcal{A}}$ be a rank $r$ central multiarrangement in $\mathbb{C}^n$ with $|{\mathcal{A}}|=m$. Define … Here $\Omega_n$ is the superspace algebra, $\lambda_L$ is the linear…

Let $M$ be a binary matroid, let $T(M;z,z)$ be its diagonal Tutte polynomial, and let the bicycle space of $M$ have dimension $d$. For an integer $k$ with … write…

Bohn–Cameron–Müller's conjecture. The group $G_{K,y}(T_M)$ is maximal, i.e. it is isomorphic to the symmetric group on $\operatorname{r}(M)$ letters:

Reiner–Smith's conjecture. If $T<T'$ in the Csikvári poset, then

Let $T$ and $T'$ be trees, and let $T<T'$ in Csikvári's poset on trees. For a tree $S$, write $T_{\operatorname{Cone}(S)}(1,y)$ for the Tutte polynomial of the cone over $S$, evalu…

Let $g$ be a positive integer, and let $\{M_i\}$ and $\{N_i\}$ be sequences of matroids. One-step separation conjecture. There exist two infinite sequences of matroids $\{M_i\}$ an…

Let $M$ be a non-separable matroid. Let $C$ be a largest circuit of $M$, let $D$ be a largest cocircuit of $M$, and let $\mathcal{B}(M)$ denote the set appearing in the genus bound…

Let $\mathcal H$ be a hypergraph and let $h$ be a hypertree. Define the order $<^{\mathrm{violet'}}_h$ from the tour of a violet Jaeger tree, and use this order to define embedding…

Let an activity notion for graphs be given, and suppose that it yields a Crapo-decomposition. Courtiel's conjecture. This activity notion fits into the framework of $\Delta$-activi…

Let $\mathcal H$ be a hypergraph, and let $\mathcal{T}_{\mathcal H}(x,y)$ denote its two-variable hypergraph Tutte polynomial. Consider the alternative embedding activities propose…

Let $T(K_n;a,b)$ denote the Tutte polynomial of the complete graph $K_n$, and let $p$ be an odd prime and $k$ a positive integer. Let $u$ denote congruence modulo the indicated po…

Tutte growth-rate existence conjecture. This limit exists if $x,y>0$. If the graphs $G_n$ do not contain loops or bridges, then the same existence assertion should also hold when…

Flag-geometric characteristic polynomial conjecture. One has

Cut-at-vertex exterior-polynomial conjecture. For every such $\mathcal H$, $\tilde{X}_{\mathcal H}=X_{\mathcal H}$, $\tilde{X}_{\mathcal H}$ is independent of the ribbon structure,…

Exterior-polynomial analogue conjecture. Theorem-level equality and ribbon-independence asserted for the interior polynomial, as well as the cut-at-vertex conjecture for the interi…

Cut-at-vertex Bernardi conjecture. This polynomial coincides with the interior polynomial $I_{\mathcal H}$ of $\mathcal H$.

Conjectured convolution-inverse formula. For the relevant indices $a,b,c,d$,

Face-module Hilbert-series conjecture. The preceding identity holds for every $\mathbb{Z}$-matroid.

Let $S$ be a combinatorial object with deletion and contraction operations, and let $e$ be an element of type $(i,j)$ when … Suppose that $\beta$ is defined recursively by … when…

Let $w\in S_n$. Let $DM_w$ be the diagram matroid associated to the permutation diagram of $w$, and let $\mathscr{P}_w$ be the corresponding positroid. Tutte-polynomial equality co…

Semistable Tutte polynomial positivity conjecture. One has $T_Z(Q;t,q)\in\mathbb{N}[t,q]$. The conjecture is motivated by the positivity theorem for $a_Z(Q)$, and the source sugges…

Let $G=(V_0,V_1,E)$ be a connected bipartite graph inducing hypergraphs $d4a2_0$ and $d4a2_1$ by taking the neighborhoods of the vertices in $V_0$ and $V_1$, respectively. The inte…

A lattice path matroid $M[P,Q]$ is represented by lattice paths $P$ and $Q$ from $(0,0)$ to $(m,r)$ that do not cross; its bases correspond to lattice paths staying in the region b…

Vergnas's conjecture. For all integers $p$, there is an odd integer $k$ such that

Zero-density conjecture. As $G$ ranges over all graphs, the zeros of $Z_G(q,v)$ are dense in regions (a)--(e). More precisely, for each fixed $q$ the zeros are dense in $v$, or for…