Combinatorics of permutation patterns, interlacing networks, and Schur functions

Abstract

In the first part, we study pattern avoidance and permutation statistics. For a set of patterns n and a permutation statistic st, let Fst/n ([Pi]; q) be the polynomial that counts st on the permutations avoiding all patterns in [Pi]. Suppose [Pi] contains the pattern 312. For a class of permutation statistics (including inversion and descent statistics), we give a formula that expresses Fst/n ([Pi]; q) in terms of these st-polynomials for some subblocks of the patterns in [Pi]. Using this recursive formula, we construct examples of nontrivial st-Wilf equivalences. In particular, this disproves a conjecture by Dokos, Dwyer, Johnson, Sagan, and Selsor that all inv-Wilf equivalences are trivial. The second part is motivated by the problem of giving a bijective proof of the fact that the birational RSK correspondence satisfies the octahedron recurrence. We define interlacing networks to be certain planar directed networks with a rigid structure of sources and sinks. We describe an involution that swaps paths in these networks and leads to a three-term relations among path weights, which immediately implies the octahedron recurrences. Furthermore, this involution gives some interesting identities of Schur functions generalizing identities by Fulmek-Kleber. Then we study the balanced swap graphs, which encode a class of Schur function identities obtained this way.