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Prefixes of Stanley's Catalan paths with odd returns to the $x$-axis -- standard version and skew Catalan-Stanley paths

Helmut Prodinger

Abstract

Stanley considered Dyck paths where each maximal run of down-steps to the $x$-axis has odd length; they are also enumerated by (shifted) Catalan numbers. Prefixes of these combinatorial objects are enumerated using the kernel method. A more challenging version of skew Dyck paths combined with Stanley's restriction is also considered.

Prefixes of Stanley's Catalan paths with odd returns to the $x$-axis -- standard version and skew Catalan-Stanley paths

Abstract

Stanley considered Dyck paths where each maximal run of down-steps to the -axis has odd length; they are also enumerated by (shifted) Catalan numbers. Prefixes of these combinatorial objects are enumerated using the kernel method. A more challenging version of skew Dyck paths combined with Stanley's restriction is also considered.
Paper Structure (3 sections, 41 equations, 4 figures)

This paper contains 3 sections, 41 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Prefixes of Dyck paths with only maximal down-runs to the $x$-axis of odd length
  3. Skew Dyck paths and the Stanley restriction

Figures (4)

  • Figure 1: Restricted paths à la Stanley of length 8. The maximal runs of down-steps to the $x$-axis are depicted in red.
  • Figure 2: Three layers of states, labelled $f,g,h$, in that order. The state $h_0$ is responsible for Stanley-Dyck paths, and all others to prefixes of them.
  • Figure 3: Three layers of states according to skew Dyck paths.
  • Figure 4: Graph to recognize skew Stanley-Dyck paths.