Skip to main content Accessibility help
×
Home

CATALAN'S TRAPEZOIDS

  • Shlomi Reuveni (a1)

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.

Copyright

References

Hide All
1.Bailey, D.F. (1996). Counting arrangements of 1’s and -1’s. Mathematics Magazine 69: 128.
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: R333R441.
3.Duchi, E. and Schaeffer, G.A. (2005). Combinatorial approach to jumping particles, Journal of Combinatorial Theory A, 110: 129.
4.Feller, W.An Introduction to Probability Theory and its Applications, Volume 1, 3rd ed.New York: Wiley.
5.Forder, H.G. (1961). Some problems in combinatorics. Mathematical Gazette 45: 199.
6.Frey, D.D. and Sellers, J.A. (2001). Generalization of the Catalan numbers. Fibonacci Quarterly, 39: 142148.
7.Reuveni, S., Eliazar, I. and Yechiali, U. (2011). Asymmetric inclusion process. Physical Review E 84: 041101.
8.Reuveni, S., Eliazar, I. and Yechiali, U. (2012). The asymmetric inclusion process: a showcase of complexity. Physical Review Letters 109: 020603.
9.Reuveni, S., Eliazar, I. and Yechiali, U. (2012). Limit laws for the asymmetric inclusion process. Physical Review E 86: 061133.
10.Reuveni, S., Hirschberg, O., Eliazar, I. and Yechiali, U. (2013). Occupation probabilities and fluctuations in the asymmetric simple inclusion process. arXiv:1309.2894.
11.Shapiro, L.W. (1976). A Catalan triangle. Discrete Mathematics 14: 83.
12.Thomas, K. (2008). Catalan Numbers with Applications. Oxford: Oxford University Press, ISBN 0-19-533454-X.

CATALAN'S TRAPEZOIDS

  • Shlomi Reuveni (a1)

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed