Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A Hybrid of Darboux’s Method and Singularity Analysis in Combinatorial Asymptotics. | A Hybrid of Darboux s Method and Singularity Analysis in Combinatorial Asymptotics Philippe Flajolet Eric Fusyy Xavier Gourdon Daniel Panario x and Nicolas Pouyanne Submitted Jun 17 2006 Accepted Nov 3 2006 Published Nov 13 2006 Mathematics Subject Classification 05A15 05A16 30B10 33B30 40E10 Abstract A hybrid method dedicated to asymptotic coefficient extraction in combinatorial generating functions is presented which combines Darboux s method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions this even in the case when the unit circle is a natural boundary. A prime application is to coefficients of several types of infinite product generating functions for which full asymptotic expansions involving periodic fluctuations at higher orders can be derived. Examples relative to permutations trees and polynomials over finite fields are treated in this way. Introduction A few enumerative problems of combinatorial theory lead to generating functions that are expressed as infinite products and admit the unit circle as a natural boundary. Functions with a fast growth near the unit circle are usually amenable to the saddle point method a famous example being the integer partition generating function. We consider here functions of moderate growth which are outside the scope of the saddle point method. We do Algorithms Project INRIA Rocquencourt F-78153 Le Chesnay France . yAlgorithms Project INRIA Rocquencourt F-78153 Le Chesnay France . z Algorithms Project and Dassault Systems France xgourdon@ . x Mathematics and Statistics Carleton University Ottawa K1S 5B6 Canada daniel@ Mathematiques Universite de Versailles 78035 Versailles France pouyanne@ . THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R103 1 so in the case where neither singularity analysis nor Darboux s method is .
