  The number of 123-avoiding permutation on $\{1,2,\ldots,n\}$ with a fixed
leading terms is counted by the ballot numbers. The same holds for
$132$-avoiding permutations. These results were proved by Miner and Pak using
the Robinson-Schensted-Knuth (RSK) correspondence to connect permutations with
Dyck paths. In this paper, we first provide an alternate proof of these
enumeration results via a direct counting argument. We then study the number of
pattern-avoiding permutations with a fixed prefix of length $t\geq1$,
generalizing the $t=1$ case. We find exact expressions for single and pairs of
patterns of length three as well as the pair $3412$ and $3421$. These
expressions depend on $t$, the extrema, and the order statistics. We also
define $r$-Wilf equivalence for permutations with a single fixed leading term
$r$, and classify the $r$-Wilf-equivalence classes for both classical and
vincular patterns of length three.
