Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: Some geometric probability problems involving the Eulerian numbers. | Some geometric probability problems involving the Eulerian numbers Frank Schmidt Rodica Simionf Department of Mathematics The George Washington University Washington DC 20052 simion@ Dedicated to Herb Wilf on the occasion of his sixty-fifth birthday Abstract We present several problems involving geometric probability. Each is related to the division of a simplex or cube by a family of hyperplanes. Both the classical Eulerian numbers and their analogue for the hyperoctahedral group arise in the solutions. 0. Introduction Consider the following general type of problem From a convex polytope P c Rn select a point x x1 x2 . xn at random according to a certain fixed distribution. Given a function f P R and a sequence of functions p0 p1 . pm P R satisfying p0 x p1 x . pm x and p0 x f x pm x what is the probability that pi-1 x f x pi x For example what is the probability that the average of the coordinates of x is at most 2 if x is selected uniformly at random from P 0 1 n the n-dimensional unit cube This arises upon choosing f x X1 x2 . Xn and the constant functions p0 x 0 p1 x 1 and p2 x 1. f Partially supported through NSF grant DMS-9108749. 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 4 no. 2 1997 R18 2 Here we present several problems of this type in which the selection of x is done uniformly at random and the functions f p0 p1 . pm are linear. Hence the problems can be reformulated geometrically as follows. Let H1 H2 . Hm be a sequence of affine hyperplanes in Rn. For each i let H and H be the two closed half-spaces in Rn determined by Hi. In terms of the original formulation of the problem the hyperplane Hi has equation f x pi x and H x G Rn f x pi x . If a point x is selected uniformly at random from the polytope P what is the probability that x lies in P n H__ 1 n H Since the selection of x is done uniformly at random this geometric probability can be expressed in terms of the n-dimensional volume of the region P n H _1 n H . Thus we are led to consider
