CATALAN'S TRAPEZOIDS

作者：Reuveni Shlomi^*

来源：Probability in the Engineering and Informational Sciences, 2014, 28(3): 353-361.

DOI：10.1017/S0269964814000047

Named after the French-Belgian mathematician Eugene 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.

• 出版日期2014-7