15 problems

Fix an integer $k\geq 1$. Let $p_{k,1}(2,n)$ denote the number of partitions of the rectangle $2\times n$ using the block-size restrictions indexed by $k$ and $1$ in the paper. A f…

Let $H_n^d$ denote the counting function for semi-magic $d$-dimensional hypercubes of order $n$. Nonpolynomiality conjecture. The function $H_n^d$ is not a polynomial when…

Let $M_n$, $S_n$, and $P_n$ denote the counting functions for the magic squares considered in the paper, with parameter $n$. Nonpolynomiality conjecture. The functions $M_n$, $S_n$…

Let $R_N$ be the quantity defined in the paper, let $g(N)=\sqrt{R_N/2}$, and let $N_k$ and $Q_{2k+1}$ be the sequences defined above. Write $\operatorname{PQ}_j(g)$ for the $j$-th…

Kirillov's conjecture. $f_{u,v,w}(N)$ is eventually polynomial in $N$.

Let $P$ be a lattice $d$-polytope, and let $\ell \in [d]$. Define $\operatorname{LD}_P^{\ell}(k)$ to be the number of $\ell$-dimensional slices of $P$ that contain the most lattice…

Let $k\in\mathbb{Z}_{>0}$, let $\Phi$ be a root system, let $\Phi^+$ be its set of positive roots, and let $\delta\in\Phi^+$. Write $\delta^c$ for the complement of $\delta$ in the…

Extremal quasi-polynomial degree conjecture. If $H_{d-1}(M;\mathbb{Q})$ and $Q_M^{\nu_n-i}(n)$ are non-trivial, then the degree of the quasi-polynomial $Q_M^{\nu_n-i}(n)$ is

The multi-width tetrahedron counting conjecture. There is a piecewise quasi-polynomial with four components counting lattice tetrahedra of multi-width $(w_1,w_2,w_3)$. There is one…

Roune–Woods conjecture. The function $D$ is an eventual quasi-polynomial (EQP).

The quasi-polynomiality conjecture. The monotone orbifold Hurwitz numbers satisfy

Let $a_i:bbZ_+crightarrowbbZ_+$, $1cle icle n$, be eventually positive functions in $cmathrm{EQP}$, where $F(a_1,\cdots,a_n)$ denotes the largest element of the group…

Quasi-polynomiality conjecture. If $\varphi_R$ is bounded by some polynomial, then $\varphi_R$ is eventually a quasi-polynomial.

Let $S_t$ be a parametric Presburger family, meaning a family of subsets of $\mathbb{N}^d$ definable over the natural numbers using quantifiers, boolean operations, and inequalitie…

Let $A(n_1,\dots,n_r)$ be an $m\times k$ matrix and $b(n_1,\dots,n_r)$ be a column vector of length $m$, with all entries linear functions of $n_1,\dots,n_r$ having integer coeffic…