10 problems

Let $m_\gamma({\rm\bf i},j,n)$ be the coefficients of the parametrized map series, and set … Thus $b=\gamma^{-1}-1$ is the nonorientability parameter. Nonnegative polynomiality con…

Let $M_\gamma({\rm\bf y},x,z)$ be the parametrized map series with expansion … Here $\gamma$ is the parameter interpolating the orientable and locally orientable map series. Polyno…

Let $\gamma$ be the parameter of the Jack symmetric functions and set … For maps, $b$ is conjectured to be an indeterminate associated with a combinatorial invariant of nonorientab…

The $b$-conjecture. The coefficients of $\psi(\bm{x},\bm{y},\bm{z};\beta)$ can be interpreted as weighted generating functions of graphs embedded into surfaces.

Rooted bridgeless maps are a class of combinatorial maps, and the paper considers a certain natural family of lambda terms associated with the same enumerative setting. Equinumeros…

Let $\tau$, $\mu$, and $\nu$ be partitions of $n\geq 1$. The coefficient $h_{\mu,\nu}^{\tau}(\beta)$ is defined through the power-sum expansion of the connected Jack three-point se…

Let a planar map be a graph embedded on a sphere, and consider planar maps up to isomorphism. Tutte's asymmetry conjecture. Almost all planar maps are asymmetric, meaning that thei…

Let $m$ and $n$ be positive integers satisfying … Singerman's conjecture. There is an orientably regular chiral map of type $\{m,n\}$. This conjecture predicts the existence of chi…

Let $\mathcal{X}_N$ denote the set of three-regular graphs with $2N$ vertices representing genus-zero one-face maps, and let $g$ be a fixed genus. Consider the scaled spacing distr…

Consider adjacency matrices of three-regular graphs representing one-face maps. The scaled eigenvalue spacing distribution is the distribution of eigenvalue spacings after normaliz…