  For given posets $P$ and $Q$ and an integer $n$, the generalized Tur\'an
problem for posets, asks for the maximum number of copies of $Q$ in a $P$-free
subset of the $n$-dimensional Boolean lattice, $2^{[n]}$.
  In this paper, among other results, we show the following:
  (i) For every $n\geq 5$, the maximum number of $2$-chains in a butterfly-free
subfamily of $2^{[n]}$ is $\left\lceil\frac{n}{2}\right\rceil\binom{n}{\lfloor
n/2\rfloor}$.
  (ii) For every fixed $s$, $t$ and $k$, a $K_{s,t}$-free family in $2^{[n]}$
has $O\left(n\binom{n}{\lfloor n/2\rfloor}\right)$ $k$-chains.
  (iii) For every $n\geq 3$, the maximum number of $2$-chains in an
$\textbf{N}$-free family is $\binom{n}{\lfloor n/2\rfloor}$, where $\textbf{N}$
is a poset on 4 distinct elements $\{p_1,p_2,q_1,q_2\}$ for which $p_1 < q_1$,
$p_2 < q_1$ and $p_2 < q_2$.
  (iv) We also prove exact results for the maximum number of $2$-chains in a
family that has no $5$-path and asymptotic estimates for the number of
$2$-chains in a family with no $6$-path.

Combinatorics