On an asymptotic method in enumeration

A remark on asymptotic enumeration of highest weights in tensor powers of a representation

We consider the semigroup $S$ of highest weights appearing in tensor powers $V^{otimes k}$ of a finite dimensional representation $V$ of a connected reductive group. We describe the cone generated by $S$ as the cone over the weight polytope of $V$ intersected with the positive Weyl chamber. From this we get a description for the asymptotic of the number of highest weights appearing in $V^{otime...

Developments in the Khintchine-Meinardus Probabilistic Method for Asymptotic Enumeration

A theorem of Meinardus provides asymptotics of the number of weighted partitions under certain assumptions on associated ordinary and Dirichlet generating functions. The ordinary generating functions are closely related to Euler’s generating function ∏∞ k=1 S(z k) for partitions, where S(z) = (1 − z)−1. By applying a method due to Khintchine, we extend Meinardus’ theorem to find the asymptotics...

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

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 ...

Asymptotic Enumeration Methods