Extended admissible functions and Gaussian limiting distributions

Abstract

Abstract. We consider an extension of Hayman's notion of admissibility to bivariate generating functions f (z, u) that have the property that the coefficients a nk satisfy a central limit theorem. It is shown that these admissible functions have certain closure properties. Thus, there is a large class of functions for which it is possible to check this kind of admissibility automatically. This is realized with help of a MAPLE program that is also presented. We apply this concept to various combinatorial examples.

Compositions into Powers of b: Asymptotic Enumeration and Parameters

“…Therefore, the functions T (x) and T 60 (x) have the same number of zeros inside a disk |x| < 3 2 by Rouché's theorem (0.062 > B T 60 ). This number equals one, since there is only one zero, a simple zero, of T 60 (x) with absolute value smaller than 3 2 . To find the exact position of that zero consider T 60 (x) + B T 60 I with the interval I = [−1, 1].…”

“…Infinite systems of functional equations appear quite frequently in the analysis of combinatorial problems, see for example the recent work of Drmota, Gittenberger and Morgenbesser [3]. Alas, their very general theorems are not applicable to our situation as the infinite matrix M does not represent an p -operator (one of their main requirements), due to the fact that its entries increase (and tend to ∞) along rows.…”

“…To find the exact position of that zero consider T 60 (x) + B T 60 I with the interval I = [−1, 1]. Again, using a bisection method (starting with 3 2 I ) plus interval arithmetic, we find an interval that contains x 0 . From this, we can extract correct digits of x 0 .…”

Limit laws of planar maps with prescribed vertex degrees

“…First, the asymptotic expansion (1.1) is known for Eulerian maps by Bender and Canfield [3]. Furthermore, a central limit theorem of the form (1.2) is known for all Eulerian maps (without degree restrictions) [12]. However, in the non-Eulerian case there are almost no results of this kind; there is only a one-dimensional central limit theorem for X (d) n for all planar maps [13].…”

“…Their asymptotic behaviour is derived in Section 3, first for the simpler case of bipartite maps (i.e. when D contains only even integers), then for families of maps without constraints on D. Sections 4 and 5 are devoted to the proof of the central limit theorem using analytic tools from [11,12]. Finally, in Section 6 we discuss combinatorics and asymptotics of bipartite maps on orientable surfaces of higher genus.…”