  This paper studies permutation statistics that count occurrences of patterns.
Their expected values on a product of $t$ permutations chosen randomly from
$\Gamma \subseteq S_{n}$, where $\Gamma$ is a union of conjugacy classes, are
considered. Hultman has described a method for computing such an expected
value, denoted $\mathbb{E}_{\Gamma}(s,t)$, of a statistic $s$, when $\Gamma$ is
a union of conjugacy classes of $S_{n}$. The only prerequisite is that the mean
of $s$ over the conjugacy classes is written as a linear combination of
irreducible characters of $S_{n}$. Therefore, the main focus of this article is
to express the means of pattern-counting statistics as such linear
combinations. A procedure for calculating such expressions for statistics
counting occurrences of classical and vincular patterns of length 3 is
developed, and is then used to calculate all these expressions. The results can
be used to compute $\mathbb{E}_{\Gamma}(s,t)$ for all the above statistics, and
for all functions on $S_{n}$ that are linear combinations of them.
