Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Generalized Cauchy identities, trees and multidimensional Brownian motions. Part I: bijective proof of generalized Cauchy identities ´ Piotr Sniady. | Generalized Cauchy identities trees and multidimensional Brownian motions. Part I bijective proof of generalized Cauchy identities Piotr Sniady Institute of Mathematics University of Wroclaw pl. Grunwaldzki 2 4 50-384 Wroclaw Poland Submitted Jul 3 2006 Accepted Jul 17 2006 Published Aug 3 2006 Mathematics Subject Classification 60J65 05A19 Abstract In this series of articles we study connections between combinatorics of multidimensional generalizations of the Cauchy identity and continuous objects such as multidimensional Brownian motions and Brownian bridges. In Part I of the series we present a bijective proof of the multidimensional generalizations of the Cauchy identity. Our bijection uses oriented planar trees equipped with some linear orders. 1 Introduction Since this paper constitutes the Part I of a series of articles we allow ourself to start with a longer introduction to the whole series. Toy example The goal of this series of articles is to discuss multidimensional analogues of the Cauchy identity. However before we do this and study our problem in its full generality we would like to have a brief look on the simplest case of the usual Cauchy identity. Even in this simplified setting we will be able to see some important features of the general case. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R62 1 Figure 1 A graphical representation of the sequence xi 1 i 25 1 1 1 1 1 1 . . It is also a graph of a continuous piecewise affine function X 0 25 R which is canonically associated to the sequence xi . Cauchy identity Cauchy identity states that for each nonnegative integer l 22 V f2p f2q p q p q l 7 1 where the sum runs over nonnegative integers p q. In order to give a combinatorial meaning to this identity we interpret the left-hand side of 1 as the number of sequences x1 . x2l 1 such that x1 . x2l 1 2 1 1 and x1 x2l 1 0. For each such a sequence xi we set p 0 to be the biggest integer such that x1 x2p 0 and set q
