9 problems

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…

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

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

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 $G(z)$.…