187 problems

A positive integer is practical if every smaller positive integer can be written as a sum of distinct divisors of it. Let $P(x)$ denote the number of practical numbers up to $x$. M…

Parity-refined asymptotic conjecture. The limit of $n^2(\Delta-\lambda_1(G))$ always exists. Furthermore,

Consider the six-vertex model with domain-wall boundary conditions on an $N$ by $N$ lattice in the disordered region, and let $Z_N$ be its partition function. Let $C$ and $\kappa$…

Let $L_k$ denote the minimum length of a permutation containing every permutation pattern of length $k$. Arratia's conjecture. Asymptotically, … Arratia's conjecture concerns the a…

Ratio-limit conjecture. One has

Tail asymptotics conjecture. For $k$ sufficiently large,

Let $G_k(n)$ be the $k\times n$ grid graph, with vertices denoted $[i,j]$, and let $R(k,n)$ be the electrical resistance between $[1,1]$ and $[k,n]$ when every edge has resistance…

Hitchin WKB conjecture. As $t\to\infty$, for every non-critical path $\gamma$,

Odd-genus normalized liminf conjecture.

Let $Z^{\gamma}$ be the discrete conformal map defined on the lattice, with parameter $\gamma$, and let $c(\gamma)$ be the constant appearing in its asymptotics. Asymptotic behavio…

Let $B(n-1)$ denote the number of $1$-perfect binary codes of length $n-1=2^m-1$, and let $\widetilde K_{\scriptscriptstyle LA}(n)$ be the lower-bound expression satisfying…

Let $(X_n)_{n\geqslant0}$ be the iterated Galton–Watson process with offspring distribution $(p_k)$ and thinning parameter $\vartheta$, where $p_0=0$, $p_1\neq1$, and $\vartheta=1$…

Let $M(n)$ denote the number of distinct multinomial coefficients arising from partitions of $n$. Exact-asymptotic-constant conjecture. … The paper explains that this would follow…

Let $M(n)$ denote the number of distinct multinomial coefficients arising from partitions of $n$. Asymptotic-growth conjecture. There exists a positive constant $C$ such that … Thi…

Let $Z^{\gamma}_{n,m}$ be the discrete map considered in the paper, with parameters $\gamma$ and $\alpha$, and let $c(\gamma)$ be the constant obtained from the asymptotics of…

Asymptotic corank-growth conjecture. As $k\to\infty$,

Let $a_p$ and $\alpha_p$ denote the numbers of open and closed plane curves, respectively, with $p$ self-intersections. Let $\gamma=-\frac{1+\sqrt{13}}6$. Asymptotic enumeration co…

Monochromatic NTIL asymptotics. The exact monochromatic no-three-in-line maxima satisfy

Let $\alpha$ be the middle real root of … For odd $n=2m+1$, let $L_m^{\mathrm{fat}}$ and $L_m^{\mathrm{thin}}$ denote the relaxation optima on the fat and thin colour classes, resp…

Convergence-rate conjecture. For every fixed $j$,

Let $H$ be an infection rule and let $M_H(n)$ denote its maximum running time. Polynomial-exponent limit conjecture. The limit … exists for every infection rule $H$. The paper does…

Let $L_k$ denote the minimum length of a permutation containing every permutation pattern of length $k$. Eriksson et al.'s conjecture. Asymptotically, … The cited 2007 work gives t…

Let $p(n)$ denote the partition function, let $\Delta$ be the forward difference operator $\Delta(a)(n)=a(n)-a(n-1)$, and let $n_0(r)$ be a threshold such that $\Delta^r(p)(n)>0$ f…

Leading-pole cancellation conjecture. The choice

Let $n_0$ be an integer. For every set of charges ${\boldsymbol r}$ with fewer than $n_0$ charges, assume that the functions $\phi^{({\boldsymbol r})}_{\boldsymbol\nu}(\tau,z)$ are…