Permutation Statistics and Multiple Pattern Avoidance

Abstract

For a set of permutation patterns $\Pi$, let $F_n^{\text{st}}(\Pi ,q)$ be the st-polynomial of permutations avoiding all patterns in $\Pi$. Suppose $312\in \Pi$. For a class of permutation statistics which includes inversion and descent statistics, we give a formula that expresses $F_n^{\text{st}}(\Pi ;q)$ in terms of these st-polynomials where we take some subblocks of the patterns in $\Pi$. 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.