Permutation statistics and multiple pattern avoidance

For a set of patterns $\Pi$, let $F{\begin{smallmatrix} \mathrm{st} \\ n \end{smallmatrix}} (\Pi ; q)$ be the $\mathrm{st}$-polynomial of permutations avoiding all patterns in $\Pi$. Suppose $312 \in \Pi$. For some permutation statistic $\mathrm{st}$, we give a formula that expresses $F{\begin{smallmatrix} \mathrm{st} \\ n \end{smallmatrix}} (\Pi ; q)$ in terms of these $\mathrm{st}$-polynomials where we take some subblocks of the patterns in $\Pi$. Using this formula, we construct examples of nontrivial $\mathrm{st}$-Wilf equivalences. In particular, this disproves a conjecture by Dokos, Dwyer, Johnson, Sagan, and Selsor that all inv-Wilf equivalences are trivial.

Full Text (PDF format)

Published 20 March 2015