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The height of skew Dyck paths with two variants of downsteps

Helmut Prodinger

Abstract

Recently, in the context of walks of hexagonal circle packings, interest has emerged in the family of skew Dyck paths with two variants of down-steps. These paths have steps $U, D_g, D_b, L=D_r$. Using generating functions, the kernel method and (in)finite linear systems, contributions to the (average) height and other enumerations are made. As in many similar instances, the average height is of order $\sqrt n$.

The height of skew Dyck paths with two variants of downsteps

Abstract

Recently, in the context of walks of hexagonal circle packings, interest has emerged in the family of skew Dyck paths with two variants of down-steps. These paths have steps . Using generating functions, the kernel method and (in)finite linear systems, contributions to the (average) height and other enumerations are made. As in many similar instances, the average height is of order .
Paper Structure (4 sections, 2 theorems, 55 equations, 4 figures)

This paper contains 4 sections, 2 theorems, 55 equations, 4 figures.

Table of Contents

  1. Skew Dyck paths
  2. Enumeration of skew Dyck paths with two variants of down-steps
  3. The recursions
  4. Further research

Key Result

Theorem 1

The average height of skew Dyck paths with two types of down-steps is asymptotic to where all such paths of semi-length $n$ are equally likely. This may be compared to the formula of standard skew Dyck paths (see garden) and for standard Dyck paths (not skew) The last result is classical deBrKnRi.

Figures (4)

  • Figure 1: Symbolic equation for the family of skew Dyck paths.
  • Figure 2: Symbolic equation for the family of skew Dyck paths with two variants of down-steps.
  • Figure 3: A path as studied in the paper of length 28 (=semi-length 14) and height 7.
  • Figure 4: An automaton to check the syntax of skew Dyck paths with two types of down-steps (blue resp. green); the left step is drawn in red.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2