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.



Symmetric distribution of maj over des in pattern avoidance classes
Let $w=a\_1 a\_2\cdots a\_n\in S\_n$. Set $w' =n+1-a\_n,n+1-a\_{n-1},\dots,n+1-a\_1$. Then (1) $\mathrm{des}(w)=\mathrm{des}(w')$, (2) $\mathrm{maj}(w)+\mathrm{maj}(w')=dn$, where $d=\mathrm{des}(w)$, and (3) $\mathrm{is}(w)=\mathrm{is}(w')$, where $\mathrm{is}(w)$ denotes the length of the longest increasing subsequence of $w$. This implies the suggested symmetry property for $S\_n(12\cdots p)$. Since the map $w\mapsto w'$ also preserves the length of the longest decreasing subsequence (in fact, it preserves the RSK insertion tableau), the symmetry property also holds for $S\_n(12\cdots p, q\cdots 21)$. 


