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<p>Motzkin paths consist of up-steps, down-steps, horizontal steps, never go below the <i>x</i>-axis and return to the <i>x</i>-axis. Versions where the return to the <i>x</i>-axis isn’t required are also considered. A path is peakless (valleyless) if <i>UD</i> (if <i>DU</i>) 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 <i>UD</i> resp. <i>DU</i>. The concepts are extended to so-called skew Motzkin paths. Methods are generating functions and the kernel method to compute explicit forms.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年8月16日

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  • Helmut Prodinger

摘要

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.