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Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs

Helmut Prodinger

TL;DR

The paper addresses the enumeration of prefixes of cornerless Motzkin paths and their skew variants, and their connection to bargraphs via a bijection. It develops a generating-function framework using the kernel method and automata-based recurrences to count occurrences of the subwords $UD$ and $DU$, introducing parameters $z$, $u$, $\sigma$, and $\tau$ to track end height and pattern counts. It delivers explicit closed-form generating functions and series for cornerless, peakless, valleyless, borderless, and open-ended cases, and extends these results to skew Motzkin paths with a left-step layer. The results unify several known sequences and provide exact tools for prefix enumeration of constrained Motzkin-type paths, with a clear route to asymptotic analysis via square-root singularities and endpoint statistics.

Abstract

Motzkin paths consist of up-steps, down-steps, horizontal steps, never go below the $x$-axis and return to the $x$-axis. Versions where the return to the $x$-axis isn't required are also considered. A path is peakless (valleyless) if $UD$ (if $DU$) never occurs. If it is both peakless and valleyless, it is called cornerless. Deutsch and Elizalde have linked cornerless Motzkin paths and bargraphs bijectly. Thus, instead of prefixes of bargraphs one might consider prefixes of cornerless Motzkin paths. In this paper, this is extended by counting the occurrences of $UD$ resp., $DU$. The concepts are extended to so-called skew Motzkin paths. Methods are generating functions and the kernel method to compute explicit forms.

Cornerless, peakless, valleyless Motzkin paths (regular and skew) and applications to bargraphs

TL;DR

The paper addresses the enumeration of prefixes of cornerless Motzkin paths and their skew variants, and their connection to bargraphs via a bijection. It develops a generating-function framework using the kernel method and automata-based recurrences to count occurrences of the subwords and , introducing parameters , , , and to track end height and pattern counts. It delivers explicit closed-form generating functions and series for cornerless, peakless, valleyless, borderless, and open-ended cases, and extends these results to skew Motzkin paths with a left-step layer. The results unify several known sequences and provide exact tools for prefix enumeration of constrained Motzkin-type paths, with a clear route to asymptotic analysis via square-root singularities and endpoint statistics.

Abstract

Motzkin paths consist of up-steps, down-steps, horizontal steps, never go below the -axis and return to the -axis. Versions where the return to the -axis isn't required are also considered. A path is peakless (valleyless) if (if ) never occurs. If it is both peakless and valleyless, it is called cornerless. Deutsch and Elizalde have linked cornerless Motzkin paths and bargraphs bijectly. Thus, instead of prefixes of bargraphs one might consider prefixes of cornerless Motzkin paths. In this paper, this is extended by counting the occurrences of resp., . The concepts are extended to so-called skew Motzkin paths. Methods are generating functions and the kernel method to compute explicit forms.
Paper Structure (4 sections, 40 equations, 11 figures)

This paper contains 4 sections, 40 equations, 11 figures.

Table of Contents

  1. Introduction
  2. The occurrences of UD and DU are counted.
  3. Enumeration of skew Motzkin paths relative to peaks and valleys
  4. Conclusion

Figures (11)

  • Figure 1: A Motzkin excursion and a Motzkin meander (ending at level $2$).
  • Figure 2: Graph (automaton) to recognize Motzkin paths
  • Figure 3: Graph (automaton) to recognize elevated Motzkin paths
  • Figure 4: Four layers of states according to the type of steps leading to them. Traditional up-steps and down-steps are black, level-steps are blue, and left steps are red.
  • Figure 5: Graph (automaton) to recognize peakless Motzkin paths; only the first few states are shown. Starting at the origin and ending at nodes labelled 0 corresponds to Motzkin paths, and ending at a node labelled $k$ to a path that ends at level $k$.
  • ...and 6 more figures