20 problems

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…

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

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…

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…

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…

For each of the three random labeled chip-firing protocols described in the source—uniformly choosing a legal firing move, uniformly choosing an unstable vertex and then a pair of…

Let $n=2m+1$, and let $\widetilde{\Delta^n}$ denote the stable configurations reachable from $\Delta^n$ by chip-firing. In the odd case, these configurations may be identified with…

Generalized parallel-loop sorting conjecture. Then $r(\mathbb{Z}\langle S\rangle)$ sorts $\Delta^{rn}$ for each $r\geq1$. This simultaneously generalizes the looped-path and parall…

For $r\geq1$, let $r\mathbb{Z}$ be the graph obtained from the infinite path $\mathbb{Z}$ by replacing each edge with $r$ parallel edges. Parallel-edge path sorting conjecture. For…

Looped-path sorting conjecture. Then $\mathbb{Z}\langle S\rangle$ sorts $\Delta^n$. Here, sorting means that every labeled stabilization of the initial configuration $\Delta^n$ is…

Tightness conjecture. Every finite graph and initial configuration is tight.

Cartwright's Picard-group conjecture.

Sink-independence conjecture. The sequence \left(\textbf{\texttt{\sum}}\,_{G,s}(B)\right)_{B\in\mathcal{C}/\sim} is independent of the choice of $s$, up to a permutation of its en…

Let $G$ be an arbitrary graph with $n$ vertices, and consider the configuration $(c,0,0,\dots,0)$, with $c$ chips on one vertex and zero chips on the remaining vertices. Reachable-…