对称格路与恒等式

doi:10.6040/j.issn.1671-9352.0.2014.196

山东大学学报（理学版） ›› 2015, Vol. 50 ›› Issue (04): 82-89.doi: 10.6040/j.issn.1671-9352.0.2014.196

对称格路与恒等式

邓利华¹, 邓玉平², Louis W. Shapiro³

1. 河南理工大学数学与信息科学学院, 河南焦作 454000;
2. 大连理工大学数学科学学院, 辽宁大连 116024;
3. 霍华德大学数学系, 美国华盛顿哥伦比亚特区 20059

收稿日期:2014-05-04 修回日期:2014-11-17 出版日期:2015-04-20 发布日期:2015-04-17
作者简介:邓利华(1983-),女,硕士,讲师,研究方向为组合数学.E-mail:denglihua@hpu.edu.cn
基金资助:
河南理工大学青年基金(Q2013-02A);国家自然科学基金资助项目(11992132)

The Riordan group and symmetric lattice paths

DENG Li-hua¹, DENG Yu-ping², Louis W. Shapiro³

1. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454000, Henan, China;
2. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, Liaoning, China;
3. Department of Mathematics, Howard University, Washington, DC 20059, USA

Received:2014-05-04 Revised:2014-11-17 Online:2015-04-20 Published:2015-04-17

摘要/Abstract

摘要： 对一种简单而又重要的组合结构——对称格路进行了研究.记d_n, m_n, s_n分别为长2n的对称Dyck格路, Motzkin格路, Schröder格路的个数.利用Riordan阵理论得到了他们之间所满足的六个组合恒等式并给予两个组合解释. 最后,得到了特殊 Riordan阵系数所满足的恒等式.根据某些恒等式估计长为2n的对称Dyck路平均中间高度和平均落在x轴上的点的个数.

关键词: 对称 Schröder格路, Riordan阵, 恒等式, 对称 Motzkin 格路, 对称 Dyck 格路

Abstract: The symmetric lattice paths are studied. Let d_n, m_n, and s_n denote the number of symmetric Dyck paths, symmetric Motzkin paths, and symmetric Schröder paths of length 2n, respectively. By using Riordan group methods, six identities relating d_n, m_n, and s_n are obtained and also two of them combinatorial proofs are given. Finally, some relations satisfied by the generic element of some special Riordan arrays are investigated and the average mid-height and the average number of points on the x-axis of symmetric Dyck paths of length 2n are obtained.

Key words: Riordan group, combinatorial identities, symmetric Motzkin paths, symmetric Dyck paths, symmetric Schröder paths

中图分类号:

O157.1

邓利华, 邓玉平, Louis W. Shapiro. 对称格路与恒等式[J]. 山东大学学报（理学版）, 2015, 50(04): 82-89.

DENG Li-hua, DENG Yu-ping, Louis W. Shapiro. The Riordan group and symmetric lattice paths[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(04): 82-89.

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对称格路与恒等式

The Riordan group and symmetric lattice paths

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