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### Schur positivity and labeled binary trees

**Algebra & Discrete Mathematics**

Speaker: | Vasu Tewari, University of Washington |

Related Webpage: | http://www.math.washington.edu/~vasut/ |

Location: | 1147 MSB |

Start time: | Mon, Nov 21 2016, 4:10PM |

Gessel introduced a multivariate formal power series tracking the distribution of ascents and descents in labeled binary

trees. In addition to showing that it was a symmetric function, he conjectured it was Schur positive in fact.

In this talk, I will present two proofs of this conjecture, the first of which utilizes a variant of a beautiful

bijection of Préville-Ratelle and Viennot concerning extensions of Tamari lattices, while the other involves solving a

functional equation in noncommutative variables. I will subsequently discuss connections to hyperplane arrangements, in

particular the Linial arrangement, by demonstrating a hidden symmetric group action on the regions of the Linial

arrangement. This uses a bijection found recently by Bernardi. Additionally, I will discuss gamma-positivity of the

coefficients of a polynomial considered by Postnikov in his work on alternating trees. Finally, time permitting, I will discuss

connections with the representation theory of the 0-Hecke algebra.

This is joint work with Ira Gessel and Sean Griffin.