Permutation Statistics and Multiple Pattern Avoidance

For a set of permutation patterns $\mathrm{\Pi }$, let $F_n^{\text{st}}(\mathrm{\Pi },q)$ be the st-polynomial of permutations avoiding all patterns in $\mathrm{\Pi }$. Suppose $312\in \mathrm{\Pi }$. For a class of permutation statistics which includes inversion and descent statistics, we give a formula that expresses $F_n^{\text{st}}(\mathrm{\Pi };q)$ in terms of these st-polynomials where we take some subblocks of the patterns in $\mathrm{\Pi }$. Using this formula, we can construct many examples of nontrivial st-Wilf equivalences. In particular, this disproves a conjecture by Dokos, Dwyer, Johnson, Sagan, and Selsor that all $\text{inv}$-Wilf equivalences are trivial.