2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/100963In this series of articles we study connections between combinatorics of multidimensional generalizations of 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 multidimensional generalizations of the Cauchy identity. Our bijection uses oriented planar trees equipped with some linear orders.Version 2: numerous misprints were corrected. Version 3: bijections described in an algorithmic wayCombinatoricsProbability60J65; 05A19text