Séminaire Lotharingien de Combinatoire, 87B.7 (2023), 36 pp.

Sergi Elizalde

Counting lattice paths by crossings and major index II: tracking descents via two-rowed arrays

Abstract. We present refined enumeration formulas for lattice paths in Z2 with two kinds of steps, by keeping track of the number of descents (i.e., turns in a given direction), the major index (i.e., the sum of the positions of the descents), and the number of crossings. One formula considers crossings between a path and a fixed line; the other considers crossings between two paths. Building on the first paper of the series, which used lattice path bijections to give the enumeration with respect to major index and crossings, we obtain a refinement that keeps track of the number of descents. The proof is based on new bijections which rely on certain two-rowed arrays that were introduced by Krattenthaler.

Received: December 13, 2021. Revised: August 3, 2022. Published: August 31, 2023.

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If you need to cite this article, here is a bibtex:

@article{Elizalde23,
author = {Elizalde, Sergi},
title = {Counting lattice paths by crossings and major index~{II}: tracking descents via two-rowed arrays},
fjournal = {S{\'e}minaire Lotharingien de Combinatoire},
journal = {S{\'e}min. Lothar. Combin.},
issn={1286-4889},
volume = {87B},
number= {7},
pages = {1--36},
year = {2023},
language = {English},
doi={10.48550/arXiv.2112.05696},
}