ILL 1983 Problem 25

How many permutations $a_1, a_2, \ldots, a_n$ of $\{1, 2, . . ., n \}$ are sorted into increasing order by at most three repetitions of the following operation: Move from left to right and interchange $a_i$ and $a_{i+1}$ whenever $a_i > a_{i+1}$ for $i$ running from $1$ up to $n - 1 \ ?$