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Dispersed Dyck paths revisited

Helmut Prodinger

Abstract

Dispersed Dyck paths are Dyck paths, with possible flat steps on level 0. We revisit and augment questions about them from the Encyclopedia of Integer Sequences, in a systematic way that uses generating functions and the kernel method.

Dispersed Dyck paths revisited

Abstract

Dispersed Dyck paths are Dyck paths, with possible flat steps on level 0. We revisit and augment questions about them from the Encyclopedia of Integer Sequences, in a systematic way that uses generating functions and the kernel method.
Paper Structure (5 sections, 51 equations, 4 figures)

This paper contains 5 sections, 51 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Counting 1-ascents
  3. Counting 1-descents
  4. Counting valleys on level 0
  5. Counting occurrences of $UUDD$

Figures (4)

  • Figure 1: Three layers of states, labelled $f,g,h$, in that order. The state $f_0$ is responsible for dispersed Dyck paths, and all others to prefixes of them.
  • Figure 2: Three layers of states, labelled $f,g,h$, in that order.
  • Figure 3: Two layers of states, labelled $f,g$, in that order.
  • Figure 4: Four layers of states, labelled $f,g,h,k$, in that order.