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For a permutation $\pi$ the major index of $\pi$ is the sum of all indices$i$ such that $\pi_i > \pi_{i+1}$. It is well known that the major index isequidistributed with the number of inversions over all permutations of length$n$. In this paper, we study the distribution of the major index overpattern-avoiding permutations of length $n$. We focus on the number$M_n^m(\Pi)$ of permutations of length $n$ with major index $m$ and avoidingthe set of patterns $\Pi$.  First we are able to show that for a singleton set $\Pi = \{\sigma\}$ otherthan some trivial cases, the values $M_n^m(\Pi)$ are monotonic in the sensethat $M_n^m(\Pi) \leq M_{n+1}^m(\Pi)$. Our main result is a study of theasymptotic behaviour of $M_n^m(\Pi)$ as $n$ goes to infinity. We prove that forevery fixed $m$ and $\Pi$ and $n$ large enough, $M_n^m(\Pi)$ is equal to apolynomial in $n$ and moreover, we are able to determine the degrees of thesepolynomials for many sets of patterns.

Major index distribution over permutation classes