Combinatorial investigation of q-analogs of two recursions for Catalan numbers with alternating signs

带交替符号的加泰罗尼亚数的两个递归的 q 类似物的组合研究

• 批准号: 237373121
• 负责人: Dr. Yu Jin
• 金额: --
• 依托单位: Institut für Diskrete Mathematik und Geometrie
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2013
• 资助国家: 德国
• 起止时间: 2012-12-31 至 2015-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/237373121?language=en
• 关键词: Combinatorial; investigation; analogs; two; recursions

Recently Zeilberger conjectured that if the integers $\{A_n\}_{n\ge 1}$ are inductively defined via the recursion for Catalan numbers with alternating signs, then the integers $\{A_n\}_{n\ge 2}$ are increasing and positive. Motivated by his conjecture and Andrews' open questions on Koshy's identity, we are primarily concerned with $q$-analogs of two recursions for Catalan numbers with alternating signs. Here, the $q$-analog of a sequence of numbers resp.\ an identity is an extension of those by parameter $q$ that naturally reduces to the original in the limit $q\rightarrow 1$. Our main objective is to generalize Zeilberger's conjecture on sequence $\{A_n\}_{n\ge 1}$ into a combinatorial and probabilistic study of $q$-analogs of $\{A_n\}_{n\ge 1}$. By further analyzing the $q$-analogs of a similar recursion for Fuss-Catalan numbers, we aim toshow the universal positivity of the coefficients of the $q$-analogs. Much more are beyond Zeilberger's conjecture. Indeed our investigation will contribute to approach the intrinsic connection of two seemingly unrelated $q$-series with positive coefficients. One is the $q$-analog of $\{A_n\}_{n\ge 1}$, the other one comes from the $q$-analog of Koshy's identity due to Andrews. Along our way we also expect to achieve some intermediate results of relevance, including completely solving Andrew's open problems on the $q$-analog of Koshy's identity in the context of the $q$-hypergeometric series, a combinatorial proof of the $q$-analog of Koshy's identity in terms of Narayana polynomials, providing a new proof of Zeilberger's conjecture by employing analytic combinatorial tools and answering Lassalle's open questions on the positivity of Schur functions associated to the complete functions generalized from the exponential generating function of Catalan numbers.

Recently Zeilberger conjectured that if the integers $\{A_n\}_{n\ge 1}$ are inductively defined via the recursion for Catalan numbers with alternating signs, then the integers $\{A_n\}_{n\ge 2}$ are increasing and positive. Motivated by his conjecture and Andrews' open questions on Koshy's identity, we are primarily concerned with $q$-analogs of two recursions for Catalan numbers with alternating signs. Here, the $q$-analog of a sequence of numbers resp.\ an identity is an extension of those by parameter $q$ that naturally reduces to the original in the limit $q\rightarrow 1$. Our main objective is to generalize Zeilberger's conjecture on sequence $\{A_n\}_{n\ge 1}$ into a combinatorial and probabilistic study of $q$-analogs of $\{A_n\}_{n\ge 1}$. By further analyzing the $q$-analogs of a similar recursion for Fuss-Catalan numbers, we aim toshow the universal positivity of the coefficients of the $q$-analogs. Much more are beyond Zeilberger's conjecture. Indeed our investigation will contribute to approach the intrinsic connection of two seemingly unrelated $q$-series with positive coefficients. One is the $q$-analog of $\{A_n\}_{n\ge 1}$, the other one comes from the $q$-analog of Koshy's identity due to Andrews. Along our way we also expect to achieve some intermediate results of relevance, including completely solving Andrew's open problems on the $q$-analog of Koshy's identity in the context of the $q$-hypergeometric series, a combinatorial proof of the $q$-analog of Koshy's identity in terms of Narayana polynomials, providing a new proof of Zeilberger's conjecture by employing analytic combinatorial tools and answering Lassalle's open questions on the positivity of Schur functions associated to the complete functions generalized from the exponential generating function of Catalan numbers.

项目成果

Equality of Shapley value and fair proportion index in phylogenetic trees

• DOI: 10.1007/s00285-014-0853-0
• 发表时间: 2015-11-01
• 期刊: JOURNAL OF MATHEMATICAL BIOLOGY
• 影响因子: 1.9
• 作者: Fuchs, Michael;Jin, Emma Yu
• 通讯作者: Jin, Emma Yu

An Asymptotic Analysis of Labeled and Unlabeled k-Trees

• DOI: 10.1007/s00453-015-0039-1
• 发表时间: 2016-08
• 期刊: Algorithmica
• 影响因子: 1.1
• 作者: M. Drmota;E. Y. Jin

Heaps and two exponential structures

堆和两个指数结构

• DOI: 10.1016/j.ejc.2015.12.007
• 发表时间: 2016
• 期刊: Eur. J. Comb.
• 作者: Emma Yu Jin

New proofs of two $q$-analogues of Koshy's formula

• DOI: 10.1090/proc/12627
• 发表时间: 2013-09
• 作者: E. Y. Jin;M. Nebel