Random k-dimensional orders: Width and number of linear extensions

Abstract

We consider the width W k (n) and number L k (n) of linear extensions of a random k-dimensional order P k (n). We show that, for each fixed k, almost surely W k (n) lies between (√k/2−C)n 1−1/k and 4kn 1-1/k , for some constant C, and L k (n) lies between (e -2 n 1-1/k ) n and (2kn 1-1/k ) n . The bounds given also apply to the expectations of the corresponding random variables. We also show that W k (n) and log L k (n) are sharply concentrated about their means.