q-Super Catalan numbers: Combinatorial identities, generating functions, and Narayana refinements

IF 1.3 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics Pub Date : 2025-05-23 DOI:10.1016/j.aam.2025.102911

Arthur Rodelet–Causse , Lenny Tevlin

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引用次数: 0

Abstract

We derive a number of combinatorial identities satisfied by the q-super Catalan numbers. In particular, we extend some of the known combinatorial identities (Touchard, Koshy, Reed Dawson) to the q-super Catalan numbers.

Next, we introduce some q-convolution identities involving q-central binomial and q-Catalan numbers, and derive a generating function for q-Catalan numbers.

Then we introduce Narayana-type refinements of the super Catalan numbers. We prove algebraically the γ-positivity of those refinements and give a combinatorial proof in a special case through the type B analog of noncrossing partitions. Then we introduce their natural q-analogs, prove their q-γ-positivity, and prove some identities they satisfy, generalizing identities of Kreweras [17] and Le Jen-Shoo [11]. Using yet another identity, we prove that these refinements are positive integer polynomials in q.

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q-超级加泰罗尼亚数：组合恒等式，生成函数，和Narayana细化

我们得到了一些由q-超加泰罗尼亚数满足的组合恒等式。特别地，我们将一些已知的组合恒等式（Touchard, Koshy, Reed Dawson）推广到q-超加泰罗尼亚数。其次，我们引入了一些涉及q-中心二项式和q-加泰罗尼亚数的q-卷积恒等式，并推导了q-加泰罗尼亚数的生成函数。然后我们引入了超加泰罗尼亚数的narayana型细化。我们从代数上证明了这些改进的γ-正性，并通过非交叉分区的B型模拟给出了一个特殊情况下的组合证明。然后引入它们的天然q-类似物，证明了它们的q-γ-正性，并证明了它们满足的一些恒等式，推广了Kreweras[17]和Le jen - sho[11]的恒等式。利用另一个恒等式，我们证明了这些改进是q的正整数多项式。

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来源期刊

Advances in Applied Mathematics 数学-应用数学

CiteScore

2.00

自引率

9.10%

发文量

审稿时长

85 days

期刊介绍： Interdisciplinary in its coverage, Advances in Applied Mathematics is dedicated to the publication of original and survey articles on rigorous methods and results in applied mathematics. The journal features articles on discrete mathematics, discrete probability theory, theoretical statistics, mathematical biology and bioinformatics, applied commutative algebra and algebraic geometry, convexity theory, experimental mathematics, theoretical computer science, and other areas. Emphasizing papers that represent a substantial mathematical advance in their field, the journal is an excellent source of current information for mathematicians, computer scientists, applied mathematicians, physicists, statisticians, and biologists. Over the past ten years, Advances in Applied Mathematics has published research papers written by many of the foremost mathematicians of our time.