24 problems

Let $P_n$ denote the number of permutations of $n$ whose cycle lengths are prime numbers, and let the asymptotic result in Theorem 1 have an error term measuring the difference bet…

Let $T_n$ be a uniformly random binary tree of size $n$, and consider the unordered trees represented by the fringe subtrees of $T_n$. The number of distinct such unordered trees i…

Let $d$ be an odd positive integer, let $R_d$ be the cyclic regular tournament described above, and let $\mathcal{H}$ denote the Hessian quantity arising in the smooth-point asympt…

Pemantel–Wilson asymptotic growth conjecture. For all three types of minimal singularities—smooth points of $\mathcal{V}$, multiple points, and cone points—and every…

Deformation conjecture. Such a deformation always exists whenever a nonminimal point is needed to obtain the asymptotics in a given direction.

Universality conjecture. Theorem should still hold: for every direction, there is a point $\b{z}\in\mathcal{V}$ such that integration near $\b{z}$ yields the correct asymptotics.

Let $cmathcal{H}_{\ell,n}(x)=\sum_{k=0}^{n}A(\ell,n,k)x^k/n!$, and let $\lambda_{\ell}=\ell\ln(\ell-1)-\ln(\mathcal{K}_{\ell})$, where…

Let $\mathcal{C}$ be a combinatorial class with generating function $f_\mathcal{C}(z)$ of radius of convergence $\rho<1$, and suppose … Assume $\alpha\notin-\mathbb{N}$, and that…

Asymptotic enumeration conjecture. For every $m\geq 1$, the number of reduced $((2m+1))$-historic trees with $n$ vertices is asymptotically equal to

Let $\mathbf{w}$ be a minimal transverse critical point, and let $N(\mathbf{w})$ be its normal cone. For a direction $\mathbf{r}$, the direction may lie on a facet or in the interi…

Melczer–Wilson conjecture. Zero-drift mostly symmetric models have asymptotic growth of the form $O(S({\mathbf{1}})^n n^{-(d+1)/2})$. This conjecture was stated incorrectly: the pa…

General-dimensional LU-factorization conjecture. In general dimension, $\mathcal{H}U=L$, where

For fixed $d$, let ${\mathcal{F}}_d(n)$ be the set of permutations of $\{1,\dots,n\}$ satisfying $i-d\leq\sigma(i)\leq i+1$ for every $i$, and let the number of $k$-cycles in such…

Let $w(z)=\sum_n a_nz^n$ be a power series with nonnegative coefficients satisfying $F(z,w(z))=0$. Suppose there exist $r>0$ and $s>0$ such that: for some $R>r$ and $S>s$, $F(z,w)$…

Logarithmic block-size conjecture. For every $i\in\mathbb N$,

Logarithmic compaction conjecture. On average, the compacted tree has size

Outermost-square-root singularity conjecture. The dominant singularity still comes solely from the outermost square root of $B_k$, implying

Let $F(\mathbf{z})=G(\mathbf{z})/H(\mathbf{z})$ be a rational function, and let Assumption (J2) denote the Jacobian nondegeneracy condition used in the paper's critical-point syste…

Let $F(\mathbf{z})$ be a rational function in $n$ variables, and consider the assumptions on $F$ required to apply the non-combinatorial complexity theorem, apart from the existenc…

Let $F({\mathbf z})\in\mathbb{Z}(z_1,\dots,z_n)$ be a rational function, and let (A5) denote that the Jacobian matrix of the relevant system of equations (GenSys1)–(GenSys5) has fu…

Noy's conjecture. An addable, minor-closed class is subcritical if and only if it has a planar forbidden minor.

For each $k\geq 0$, let $c_k$ be the limiting proportion of vertices of rank $k$, and write it in lowest terms. Let $denom(c_k)$ be its denominator. For $k\geq 2$, let $p_k$ be the…

Let $c_k$ be the limiting proportion of vertices of rank $k$, and let $denom(c_k)$ denote its denominator when $c_k$ is written in lowest terms. Denominator prime-divisor bound con…

Let $\mathcal{G}$ be an addable, minor-closed class of graphs, let $G(z)$ be its exponential generating function, and let $\rho$ be the radius of convergence of $G(z)$.…