(p,q,t)-Catalan continued fractions, gamma expansions and pattern avoidances

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics Pub Date : 2024-07-08 DOI:10.1016/j.aam.2024.102735

Bin Han , Qiongqiong Pan

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引用次数: 0

Abstract

We introduce a kind of $(p, q, t)$ -Catalan numbers of Type A by generalizing the J-type continued fraction formula, we prove that the corresponding expansions could be expressed by the polynomials counting permutations on $S_{n} (321)$ by various descent statistics. Moreover, we introduce a kind of $(p, q, t)$ -Catalan numbers of Type B by generalizing the J-type continued fraction formula, we prove that the Taylor coefficients and their γ-coefficients could be expressed by the polynomials counting permutations on $S_{n} (3124, 4123, 3142, 4132)$ by various descent statistics. Our methods include permutation enumeration techniques involving variations of bijections from permutation patterns to labeled Motzkin paths and modified Foata-Strehl action.

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(p,q,t)-卡塔朗续分、伽马展开和模式回避

我们通过概括 J 型续分数公式，引入了一种 A 型（p,q,t）-卡塔兰数，并证明了相应的展开式可以用 Sn(321) 上的多项式计数排列组合通过各种下降统计来表示。此外，我们通过概括 J 型续分数公式引入了一种 B 型（p,q,t）-卡塔兰数，并通过各种下降统计证明泰勒系数及其 γ 系数可以用 Sn(3124,4123,3142,4132) 上的多项式计数排列来表示。我们的方法包括包络枚举技术，涉及从包络模式到标注莫兹金路径的双射变化，以及修正的 Foata-Strehl 作用。

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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.