How to uniformly sample a sorted simplex


I am looking for an algorithm to uniformly generate a descending array of N random numbers, such that the sum of the N numbers is 1, and all numbers lie within 0 and 1. For example, N=3, the random point (x, y, z) should satisfy:

x + y + z = 1 0 <= x <= 1 0 <= y <= 1 0 <= z <= 1 x >= y >= z 

My guess is all i have to do is uniformly sample a simplex (Uniform sampling from a simplex), and then sort the elements. But i’m not sure whether the result sampling algorithm is uniform.

Also, rejection sampling is not ideal for me, because i’ll use this for high dimension.

Thanks!