It appears from computation to be the case (and would prove at least one clause of a conjecture advanced by Bruce Sagan and collaborators in a recent preprint) that in some pattern avoidance classes of permutations, the distribution of the major index is symmetric among permutations with a given descent number. For instance, $$\sum_{S_6(1234)} q^{maj(\sigma)}t^{des(\sigma)} = \dots + (10 q^6 + 35 q^7 + 66 q^8 + 80 q^9 + 66 q^{10} + 35 q^{11} + 10 q^{12}) t^3 + \dots.$$

As you can see, the coefficient of $t^3$ is a symmetric polynomial.

This is not the case for all avoidance classes: for instance, $$\sum_{S_6(2134)} q^{maj(\sigma)}t^{des(\sigma)} = \dots + (4 q^6 + 21 q^7 + 42 q^8 + 61 q^9 + 56 q^{10} + 35 q^{11} + 10 q^{12}) t^3 + \dots.$$

Before I start hammering at this, I was wondering if this was known to be the case for any particular avoidance classes, and if so, which. Since I have not even found any papers that seem to deal with the subject, pointers to one you know of would also be gratefully received.