37 problems

Let $G$ be a graph with vertex set $V(G)$, and let $\sigma$ be a position of the parallel chip-firing game on $G$. If $p(\sigma)$ denotes the period of $\sigma$, then Bitar's conje…

A configuration $F(x,y)$ consists of rows of chips, with the bottom triangle formed by the minimal rows at the bottom of the configuration. Let the longest row be a row having maxi…

Let $\Delta^{k,m}$ be the star graph with $k$ branches, each receiving $m$ labeled chips, and consider the random labeled chip-firing process in which every firable vertex is equal…

Let $N_{k,\ell}$ denote the number of labeled chips starting at the root of a $k$-ary tree in the setting of the paper, and let a stable configuration be one obtained from these ch…

Let $\ell$ be a positive integer, and consider stable configurations of $2^{\ell}-1$ labeled chips on a binary tree. For a vertex $v$ and an integer $i$, compare the $i$th smallest…

Let $k$ and $cell$ be positive integers, and consider a directed $k$-ary tree with $k^{\ell}$ labeled chips initially at the root. Let $Z_k(\ell)$ denote the specified stable confi…

Ballot-property conjecture. In a stable configuration, the whole tree and every subtree satisfy the ballot property.

Middle-stair period-two conjecture. Every parallel chip-firing game on $G(V,E)$ satisfying these inequalities has $T=2$, and hence activity $\frac{1}{2}$. The authors report that t…

Let $D$ be a directed graph, let $c$ denote the number of chips, and let $T_c(D)$ denote the relevant possible period quantity for parallel chip-firing games on $D$ with $c$ chips.…

Let $K_{a,\,b}$ be the complete bipartite graph with $a\leq b$, and let $\tau(a,\,b)$ be the maximum possible period of a parallel chip-firing game on an orientation of $K_{a,\,b}$…

Let $K_{m\times n}$ denote the king's graph on an $m\times n$ grid, with $4\leq m\leq n$. The gonality of a graph is denoted by $\operatorname{gon}(G)$. Eventual gonality conjectur…

Let a flat clusteron be an initial state consisting of $N$ violinists in consecutive occupied rooms, and let $F_{N,r}$ denote the final shadow indexed by $r$. At each state, choose…

Most likely terminal configuration conjecture. For all $n$, the most likely terminal configuration is the unique binary search tree on the complete binary tree with $2^n-1$ nodes.

Let a finite rational irreducible Markov chain have $n$ states, transition matrix with hunger-game matrix $H$, and basin of attraction in the hyperplane $Z$. The vectors…

Let a finite irreducible Markov chain have rational transition probabilities, stationary distribution $\pi$, and let $n$ be the least common denominator of the entries of $\pi$. A…

A chip configuration on the one-dimensional grid is ordered by reachability under chip-firing; the resulting poset is called the configuration poset. A lattice is distributive if i…

Consider a labeled chip-firing process on a graph in which each pair of adjacent sites is connected by $r$ edges and each site has $r$ self-loops. A firing move chooses $3r$ chips…

Let $W$ be a connected graphon and let $\sigma$ be a chip configuration on $W$. For each $n$, let $u_n(x)$ denote the number of chips fired at $x$ during the first $n$ steps of chi…

Clancy–Leake–Payne conjecture. As $n$ tends to infinity,

Truncated Ehrhart-like polynomial conjecture. For every $\lambda\in P$ and every good $\mathbf{k}\in\mathbb{N}[\Phi]^W$,

Let $Phi=B_n$. Let $P$ be the weight lattice, and let $lambda\in P$ be a connected, dominant weight. Type B connected-weight confluence conjecture. Central-firing is confluent from…

Let $Phi$ be a root system, let $Q$ be its root lattice, let $P$ be its weight lattice, let $rho$ be the Weyl vector, and let $Omega$ denote the set of fundamental weights. Let…

Consider the labeled chip-firing process on the integer line, where a firing of two chips at the same vertex moves the lesser-labeled chip one step right and the greater-labeled ch…

Let $Phi=A_{2n}$ with $n a0 1$, and let $omegan$ be the $n$th fundamental weight. Central-firing is the process on a root system in which one applies the allowed central-firing mov…

Let a configuration consist of an even number of distinguishable chips with integer labels on the integer line, initially all at the origin. A firing chooses two chips at the same…