Enumerating pattern-avoiding permutations by leading terms

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.