Permutation Statistics and Multiple Pattern Avoidance

TL;DR

This paper develops a formula relating permutation pattern avoidance st-polynomials to subblocks, enabling the construction of nontrivial st-Wilf equivalences and disproving a prior conjecture about inversion Wilf equivalences.

Contribution

It introduces a formula connecting st-polynomials for pattern-avoiding permutations with subblock patterns, revealing nontrivial equivalences and challenging existing conjectures.

Findings

Disproved the conjecture that all inv-Wilf equivalences are trivial.

Provided a formula to relate st-polynomials with subblock patterns.

Constructed examples of nontrivial st-Wilf equivalences.

Abstract

For a set of permutation patterns $\Pi$, let $F^\text{st}_n(\Pi,q)$ 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 $F^\text{st}_n(\Pi;q)$ 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.