# Expect size of $k^{th}$ layer of a POSET

Is this known?

What is the expected width of the $k^{th}$ layer (anti-chain layer) of a $d$-dimensional partially ordered set of $n$ elements formed by product of $d$ random liner orders chosen from the same distribution.

It is well known for k = 1 (http://link.springer.com/article/10.1007/BF00582738). But not able to find any reference for k > 1.