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Article
Parametric Catalan Numbers and Catalan Triangles
Linear Algebra and Its Applications
(2013)
Abstract
Here presented a generalization of Catalan numbers and Catalan triangles associated with two parameters based on the sequence characterization of Bell-type Riordan arrays. Among the generalized Catalan numbers, a class of large generalized Catalan numbers and a class of small generalized Catalan numbers are defined, which can be considered as an extension of large Schroder numbers and small Schroder numbers, respectively. Using the characterization sequences of Bell-type Riordan arrays, some properties and expressions including the Taylor expansions of generalized Catalan numbers are given. A few combinatorial interpretations of the generalized Catalan numbers are also provided. Finally, a generalized Motzkin numbers and Motzkin triangles are defined similarly. An interrelationship among parametrical Catalan triangle, Pascal triangle, and Motzkin triangle is presented based on the sequence characterization of Bell-type Riordan arrays.
Keywords
- Catalan number,
- large Schroder number,
- small Scroder number,
- Motzkin numbers,
- Riordan array,
- Bell-type Riordan array,
- characteristic sequence.
Disciplines
Publication Date
Winter February, 2013
Publisher Statement
Linear Algebra and its Applications is published by Elsevier, http://www.journals.elsevier.com/linear-algebra-and-its-applications/.
Citation Information
Tian-Xiao He. "Parametric Catalan Numbers and Catalan Triangles" Linear Algebra and Its Applications Vol. 438 Iss. 3 (2013) Available at: http://works.bepress.com/tian_xiao_he/15/