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CATALAN'S TRAPEZOIDS

Published online by Cambridge University Press:  18 March 2014

Shlomi Reuveni
Shlomi Reuveni*
Affiliation:
Department of Statistics and Operations Research, School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel; Department of Systems Biology, Harvard University, 200 Longwood Avenue, Boston, MA 02115, USA. Email: shlomireuveni@hotmail.com
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Abstract

Named after the French–Belgian mathematician Eugène Charles Catalan, Catalan's numbers arise in various combinatorial problems [12]. Catalan's triangle, a triangular array of numbers somewhat similar to Pascal's triangle, extends the combinatorial meaning of Catalan's numbers and generalizes them [1,5,11]. A need for a generalization of Catalan's triangle itself arose while conducting a probabilistic analysis of the Asymmetric Simple Inclusion Process (ASIP) — a model for a tandem array of queues with unlimited batch service [7–10]. In this paper, we introduce Catalan's trapezoids, a countable set of trapezoids whose first element is Catalan's triangle. An iterative scheme for the construction of these trapezoids is presented, and a closed-form formula for the calculation of their entries is derived. We further discuss the combinatorial interpretations and applications of Catalan's trapezoids.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 28 , Issue 3 , July 2014 , pp. 353 - 361
Copyright
Copyright © Cambridge University Press 2014 

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References

1.Bailey, D.F. (1996). Counting arrangements of 1’s and -1’s. Mathematics Magazine 69: 128.CrossRefGoogle Scholar
2.Blythe, R.A. and Evans, M.R. (2007). Nonequilibrium steady states of matrix-product form: a solvers guide. Journal of Physics A: Mathematical and Theoretical, 40: R333–R441.CrossRefGoogle Scholar
3.Duchi, E. and Schaeffer, G.A. (2005). Combinatorial approach to jumping particles, Journal of Combinatorial Theory A, 110: 1–29.CrossRefGoogle Scholar
4.Feller, W.An Introduction to Probability Theory and its Applications, Volume 1, 3rd ed.New York: Wiley.Google Scholar
5.Forder, H.G. (1961). Some problems in combinatorics. Mathematical Gazette 45: 199.CrossRefGoogle Scholar
6.Frey, D.D. and Sellers, J.A. (2001). Generalization of the Catalan numbers. Fibonacci Quarterly, 39: 142–148.Google Scholar
7.Reuveni, S., Eliazar, I. and Yechiali, U. (2011). Asymmetric inclusion process. Physical Review E 84: 041101.CrossRefGoogle ScholarPubMed
8.Reuveni, S., Eliazar, I. and Yechiali, U. (2012). The asymmetric inclusion process: a showcase of complexity. Physical Review Letters 109: 020603.CrossRefGoogle ScholarPubMed
9.Reuveni, S., Eliazar, I. and Yechiali, U. (2012). Limit laws for the asymmetric inclusion process. Physical Review E 86: 061133.CrossRefGoogle ScholarPubMed
10.Reuveni, S., Hirschberg, O., Eliazar, I. and Yechiali, U. (2013). Occupation probabilities and fluctuations in the asymmetric simple inclusion process. arXiv:1309.2894.Google Scholar
11.Shapiro, L.W. (1976). A Catalan triangle. Discrete Mathematics 14: 83.CrossRefGoogle Scholar
12.Thomas, K. (2008). Catalan Numbers with Applications. Oxford: Oxford University Press, ISBN 0-19-533454-X.Google Scholar