## Combinatorics of permutation patterns, interlacing networks, and Schur functions

##### Author(s)

Trongsiriwat, Wuttisak
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##### Other Contributors

Massachusetts Institute of Technology. Department of Mathematics.

##### Advisor

Alexander Postnikov.

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Show full item record##### 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.

##### Description

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015. Cataloged from PDF version of thesis. Includes bibliographical references (pages 71-73).

##### Date issued

2015##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Publisher

Massachusetts Institute of Technology

##### Keywords

Mathematics.