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k-non-crossing trees and edge statistics modulo k

Helmut Prodinger

Abstract

Instead of $k$-Dyck paths we consider the equivalent concept of $k$-non-crossing trees. This is our preferred approach relative to down-step statistics modulo $k$ (first studied by Heuberger, Selkirk, and Wagner by different methods). One symmetry argument about subtrees is needed and the rest goes along the lines of a paper by Flajolet and Noy.

k-non-crossing trees and edge statistics modulo k

Abstract

Instead of -Dyck paths we consider the equivalent concept of -non-crossing trees. This is our preferred approach relative to down-step statistics modulo (first studied by Heuberger, Selkirk, and Wagner by different methods). One symmetry argument about subtrees is needed and the rest goes along the lines of a paper by Flajolet and Noy.
Paper Structure (3 sections, 12 equations, 7 figures)

This paper contains 3 sections, 12 equations, 7 figures.

Table of Contents

  1. Non-crossing trees revisited
  2. An application
  3. Generalization

Figures (7)

  • Figure 1: A non-crossing trees with 10 nodes and separators indicating where the non-root nodes split into the left part and the right part.
  • Figure 2: Left resp. right edges depicted in different colors; by design, the edges emanating from the root are all right edges.
  • Figure 3: A non-crossing tree and the corresponding 2-Dyck path.
  • Figure 4: Transforming the tree. The number of green edges corresponds to the brown down-steps.
  • Figure 5: The decomposition of a 2-Dyck path.
  • ...and 2 more figures