JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2015, Vol. 50 ›› Issue (04): 82-89.doi: 10.6040/j.issn.1671-9352.0.2014.196

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The Riordan group and symmetric lattice paths

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

  1. 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

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Abstract: The symmetric lattice paths are studied. Let dn, mn, and sn 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 dn, mn, and sn 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

CLC Number: 

  • O157.1
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