  Given a finite poset P, we consider the largest size La(n,P) of a family of
subsets of $[n]:=\{1,...,n\}$ that contains no subposet P. This problem has
been studied intensively in recent years, and it is conjectured that $\pi(P):=
\lim_{n\rightarrow\infty} La(n,P)/{n choose n/2}$ exists for general posets P,
and, moreover, it is an integer. For $k\ge2$ let $\D_k$ denote the $k$-diamond
poset $\{A< B_1,...,B_k < C\}$. We study the average number of times a random
full chain meets a $P$-free family, called the Lubell function, and use it for
$P=\D_k$ to determine $\pi(\D_k)$ for infinitely many values $k$. A stubborn
open problem is to show that $\pi(\D_2)=2$; here we make progress by proving
$\pi(\D_2)\le 2 3/11$ (if it exists).
