Séminaire Lotharingien de Combinatoire, 78B.73 (2017), 12 pp.
Ira M. Gessel, Sean Griffin and Vasu Tewari
Schur Positivity and Labeled Binary Trees
Abstract.
The first author introduced a multivariate generating function that
tracks the distribution of ascents and descents on labeled plane
binary trees and conjectured that it was Schur positive. In this
article, we give a sketch for a proof of the stronger statement that
the generating function restricted to trees with a fixed canopy is
Schur positive. Central to our approach is a weighted extension of a
bijection of Préville-Ratelle and Viennot relating pairs of paths and
binary trees. We apply our results to construct a
Sn-action on the regions of the Linial arrangement using
a bijection of Bernardi. We then establish the γ-positivity for
the distribution of right descents over local binary search trees.
Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.
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