Geodesic
Schur Positivity and Labeled Binary Trees
Séminaire lotharingien de combinatoire, 78B (2017)

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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. 

@article{SLC_2017_78B_a72,
     author = {Ira M. Gessel and Sean Griffin and Vasu Tewari},
     title = {Schur {Positivity} and {Labeled} {Binary} {Trees}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {78B},
     year = {2017},
     url = {http://geodesic.mathdoc.fr/item/SLC_2017_78B_a72/}
}
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VL  - 78B
PB  - mathdoc
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%A Vasu Tewari
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%V 78B
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%F SLC_2017_78B_a72
Ira M. Gessel; Sean Griffin; Vasu Tewari. Schur Positivity and Labeled Binary Trees. Séminaire lotharingien de combinatoire, 78B (2017). http://geodesic.mathdoc.fr/item/SLC_2017_78B_a72/