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Báo cáo toán học: "Asymptotics of the average height of 2–watermelons with a wall"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Asymptotics of the average height of 2–watermelons with a wall. | Asymptotics of the average height of 2-watermelons with a wall. Markus Fulmek Fakultat far Mathematik Universitat Wien Nordbergstrafie 15 A-1090 Wien Austria Markus.Fulmek@Univie.Ac.At Submitted Jan 10 2007 Accepted Sep 3 2007 Published Sep 7 2007 Mathematics Subject Classification 05A16 Abstract We generalize the classical work of de Bruijn Knuth and Rice giving the asymptotics of the average height of Dyck paths of length n to the case of p-watermelons with a wall i.e. to a certain family of p nonintersecting Dyck paths simple Dyck paths being the special case p 1. An exact enumeration formula for the average height is easily obtained by standard methods and well-known results. However straightforwardly computing the asymptotics turns out to be quite complicated. Therefore we work out the details only for the simple case p 2. 1 Introduction The model of vicious walkers was originally introduced by Fisher 10 and received much interest since it leads to challenging enumerative questions. Here we consider special configurations of vicious walkers called p-watermelons with a wall. Briefly stated a p-watermelon of length n is a family Pl . .Pp of p nonintersecting lattice paths in Z2 where Pị starts at 0 2i 2 and ends at 2n 2i 2 for i 1 . p all the steps are directed north-east or south-east i.e. lead from lattice point i j to i 1 j 1 or to i 1 j - 1 Research supported by the National Research Network Analytic Combinatorics and Probabilistic Number Theory funded by the Austrian Science Foundation. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R64 1 Figure 1 A 6-watermelon of length 46 and height 20 level 20 no two paths Pi Pj have a point in common this is the meaning of nonintersecting . The height of a p-watermelon is the y-coordinate of the highest lattice point contained in any of its paths since the paths are nonintersecting it suffices to consider the lattice points contained in the highest path Pp see Figure 1 for an illustration. A p-watermelon of length n .