Skip to main content Accessibility help
×
Home
Hostname: page-component-6c8bd87754-5dxdz Total loading time: 0.269 Render date: 2022-01-17T01:49:38.572Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

CATALAN'S TRAPEZOIDS

Published online by Cambridge University Press:  18 March 2014

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

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
Copyright
Copyright © Cambridge University Press 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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: R333R441.CrossRefGoogle Scholar
3.Duchi, E. and Schaeffer, G.A. (2005). Combinatorial approach to jumping particles, Journal of Combinatorial Theory A, 110: 129.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: 142148.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
7
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

CATALAN'S TRAPEZOIDS
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

CATALAN'S TRAPEZOIDS
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

CATALAN'S TRAPEZOIDS
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *