Alice L. L. Gao ; Emily X. L. Gao ; Patxi Laborde-Zubieta ; Brian Y. Sun - Enumeration of Corners in Tree-like Tableaux

dmtcs:1408 - Discrete Mathematics & Theoretical Computer Science, December 2, 2016, Vol. 18 no. 3 -
Enumeration of Corners in Tree-like TableauxArticle

Authors: Alice L. L. Gao ; Emily X. L. Gao ; Patxi Laborde-Zubieta ; Brian Y. Sun

    In this paper, we confirm conjectures of Laborde-Zubieta on the enumeration of corners in tree-like tableaux and in symmetric tree-like tableaux. In the process, we also enumerate corners in (type $B$) permutation tableaux and (symmetric) alternative tableaux. The proof is based on Corteel and Nadeau's bijection between permutation tableaux and permutations. It allows us to interpret the number of corners as a statistic over permutations that is easier to count. The type $B$ case uses the bijection of Corteel and Kim between type $B$ permutation tableaux and signed permutations. Moreover, we give a bijection between corners and runs of size 1 in permutations, which gives an alternative proof of the enumeration of corners. Finally, we introduce conjectural polynomial analogues of these enumerations, and explain the implications on the PASEP.

    Volume: Vol. 18 no. 3
    Section: Combinatorics
    Published on: December 2, 2016
    Accepted on: November 9, 2016
    Submitted on: November 28, 2016
    Keywords: Mathematics - Combinatorics

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