Combinatorics of the zeta map on rational Dyck paths

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:141

Article

Ceballos, Cesar¹  Denton, Tom²  Hanusa, Christopher R. H.³ 

[1] Univ Vienna, Fac Math, Oskar Morgenstern Pl 1, A-1090 Vienna, Austria

[2] York Univ, Dept Math & Stat, Toronto, ON M3J 1P3, Canada

[3] CUNY Queens Coll, Dept Math, 65-30 Kissena Blvd, Flushing, NY 11367 USA

关键词: Dyck path;    Rational Dyck path;    Lattice path;    Core partition;    Zeta map;    Eta map;    Sweep map;    Lasers Conjugate-area map;    Area statistic;   

DOI  :  10.1016/j.jcta.2016.02.002

来源: Elsevier

【 摘 要 】

An (a, b)-Dyck path P is a lattice path from (0,0) to (b, a) that stays above the line y = fx. The zeta map is a curious rule that maps the set of (a, b) -Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of P and zeta of P conjugate is enough to recover P. Our method begets an area-preserving involution x on the set of (a, b) -Dyck paths when C is a bijection, as well as a new method for calculating C-1 on classical Dyck paths. For certain nice (a, b) -Dyck paths we give an explicit formula for C-1 and x and for additional (a, b) -Dyck paths we discuss how to compute C-1 and x inductively. We also explore Armstrong's skew length statistic and present two new combinatorial methods for calculating the zeta map involving lasers and interval intersections. We provide a combinatorial statistic 5 that can be used to recursively compute zeta(-1) and show that 5 is computable from C(P) hi the Fuss-Catalan case. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free