Asymptotic enumeration of graphical regular representations

Distinguished positive regular representations

Let $G$ be a tamely ramified reductive $p$-adic‎ ‎group‎. ‎We study distinction of a class of irreducible admissible representations‎ ‎of $G$ by the group of fixed points $H$ of an involution‎ ‎of $G$‎. ‎The representations correspond to $G$-conjugacy classes of‎ ‎pairs $(T,phi)$‎, ‎where $T$ is a‎ ‎tamely ramified maximal torus of $G$ and $phi$ is a quasicharacter‎ ‎of $T$ whose restriction t...

On quasiabelian Cayley graphs and graphical doubly regular representations

Asymptotic enumeration of orientations

We find the asymptotic number of 2-orientations of quadrangulations with n inner faces, and of 3orientations of triangulations with n inner vertices. We also find the asymptotic number of prime 2-orientations (no separating quadrangle) and prime 3-orientations (no separating triangle). The estimates we find are of the form c ·n−αγn, for suitable constants c, α, γ, with α = 4 for 2-orientations ...

An Enumeration of Graphical Designs

Let Ψ(t, k) denote the set of pairs (v, λ) for which there exists a graphical t-(v, k, λ) design. Most results on graphical designs have gone to show the finiteness of Ψ(t, k) when t and k satisfy certain conditions. The exact determination of Ψ(t, k) for specified t and k is a hard problem and only Ψ(2, 3), Ψ(2, 4), Ψ(3, 4), Ψ(4, 5), and Ψ(5, 6) have been determined. In this paper, we determin...

Asymptotic Enumeration Methods

12 Large singularities of analytic functions 113 12.1 The saddle point 13 Multivariate generating functions 128 14 Mellin and other integral transforms 134 15 Functional equations, recurrences, and combinations of methods 137 15.1 Implicit functions, graphical enumeration, and related