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<p>We introduce a family of generalized Narayana polynomials and their associated generalized Catalan numbers. A combinatorial description based on Dyck paths with coloured ascents is provided for these polynomials and the associated numbers. Next, we derive a convolution formula of the generating function for the generalized Narayana polynomials, which generalizes the convolution identity of the Catalan numbers. Furthermore, we present several identities involving the generalized Narayana polynomials under special conditions, which provide a unified formula for many classical combinatorial numbers.</p>

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Generalized Narayana Polynomials and Dyck Paths with Coloured Ascents

  • 期刊论文

  • 发表日期：

    2025年9月23日

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  • Hua Xin,
  • Huan Xiong

摘要

We introduce a family of generalized Narayana polynomials and their associated generalized Catalan numbers. A combinatorial description based on Dyck paths with coloured ascents is provided for these polynomials and the associated numbers. Next, we derive a convolution formula of the generating function for the generalized Narayana polynomials, which generalizes the convolution identity of the Catalan numbers. Furthermore, we present several identities involving the generalized Narayana polynomials under special conditions, which provide a unified formula for many classical combinatorial numbers.