12 problems

Let $\mathbb{F}_n$ be a Hirzebruch surface, and consider its double tropical Welschinger invariants as functions of the relevant discrete geometric data. Ardila–Brugallé's conjectu…

Quasipolynomial divisibility conjecture. For every fixed tableau $T$, there exists a quasipolynomial in $n$ such that, for $n$ sufficiently large, the orbit length of $T[n]$ always…

Let $i\in\mathbb{Z}_{\geq 2}$ be the number of interior lattice points of a half-integral polygon, and suppose the polygon has at least $3$ boundary lattice points. Enumeration con…

For $n,k\in\mathbb{N}$, modulus $N\in\mathbb{N}$, and residue $R\in\mathbb{Z}/N\mathbb{Z}$, define … Here $[q^{\alpha}]\left[{n \atop k}\right]_q$ denotes the coefficient of…

Let $u_{\mathbb P}(q;n)$ be the quasipolynomial for a piece $\mathbb P$, and write $p_i$ for the period of its coefficient $\gamma_i$. Coefficient-period monotonicity conjecture. T…

For a piece $\mathbb P$ with move set $\mathbf{M}$, write … For a positive integer $\lambda$, let $\mathbb P^{\operatorname{lcm}}_\lambda$ be the lcm-maximal piece whose move set c…

Let $\mathbf{M}$ be the move set of a piece $\mathbb P$, with basic moves $(c_r,d_r)$, and set … Let $\gamma_4$ denote the fourth coefficient of the quasipolynomial…

Let $\mathbf{M}$ be the move set of a piece $\mathbb P$, with basic moves $(c_r,d_r)$, and write … Let $\gamma_3$ denote the third coefficient of the quasipolynomial…

Maximal-move-range piece conjecture. Among all pieces with $\|\mathbb P\|\leq\lambda$, $\mathbb P^{\max}_\lambda$ maximizes the period of every coefficient $\gamma_i$, for fixed…

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-…

Let $G$ be a finite connected graph with $n$ vertices, and let $(c,0,0,\dots,0)$ be the configuration having $c$ chips on one vertex and none on the others. Reachability quasipolyn…

Period and variation conjectures. The following assertions are proposed: