30 problems

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…

Profile quasi-polynomial conjecture. If $R$ has bounded signature or finite kernel and $\varphi_R$ is bounded by some polynomial, then $\varphi_R$ is a quasi-polynomial.

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 $N$.

Fix an integer $k\geq1$, and let $p_{k,1}(2,n)$ be the restricted rectangle-partition function from the paper. For each $j$, write $[r_{i,j,0},r_{i,j,1},\ldots,r_{i,j,j-1}]_j$ for…

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 $P$ 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…

Chaiken–Hanusa–Zaslavsky's conjecture. Every quasi-polynomial for the anassa has $(m)_k$ as a factor.

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…

Generalized Kronecker stability conjecture. The sequence $b_d$ is a quasi-polynomial in $d$ for all sufficiently large $d$. The ordinary generalized stability family is already kno…

Stretching quasi-polynomial conjecture. If $\lambda_2$ and $\beta_2$ are both even, then $\mathrm{mult}_{s\lambda}(s\beta)$ is a polynomial in $s$ when $\kappa_1$ is even and, exce…

Let $P$ be a $d$-dimensional rational polytope in $\mathbb{R}^m$. For a function of quasi-polynomial type, its grade is the smallest integer $\delta\geq -1$ such that the coefficie…

Let $\Lambda$ be a lattice, let ${\mathcal{S}}(\Lambda)$ be the space of piecewise quasi-polynomial functions, and let ${\mathcal{A}}$ be the asymptotic expansion map. Let…

Woods's conjecture. Every parametric Presburger family exhibits eventual quasi-polynomial behavior.

Let $\\{S_t:t\in\\N\\}$ be a parametric Presburger family, and suppose $S_t$ is finite. Write $\max(S_t)$ for its maximum. Woods's parametric Frobenius conjecture. The function…

Let $t$ range over $\\N$ and fix $d \in \\N$. A parametric Presburger family is a family $\\{S_t:t\in\\N\\}$ of subsets of $\\Z^d$ defined by an inequality … with…

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.

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

Fix a positive integer $a$. Let $M_{g,n}^{[a]}(b_1,\ldots,b_n)$ be the enumeration numbers of $a$-hypermaps. A function is a quasi-polynomial modulo $a$ if its polynomial expressio…