Recursively-Defined Combinatorial Functions: The Case of Binomial and Multinomial Coefficients and Probabilities
Ali Muhammad Ali Rushdi *
Department of Electrical and Computer Engineering, King Abdulaziz University, P.O.Box 80204, Jeddah, 21589, Kingdom of Saudi Arabia.
Mohamed AbdulRahman Al-Amoudi
Department of Electrical and Computer Engineering, King Abdulaziz University, P.O.Box 80204, Jeddah, 21589, Kingdom of Saudi Arabia.
*Author to whom correspondence should be addressed.
Abstract
This paper studies a prominent class of recursively-defined combinatorial functions, namely, the binomial and multinomial coefficients and probabilities. The paper reviews the basic notions and mathematical definitions of these four functions. Subsequently, it characterizes each of these functions via a recursive relation that is valid over a certain two-dimensional or multi-dimensional region and is supplemented with certain boundary conditions. Visual interpretations of these characterizations are given in terms of regular acyclic signal flow graphs. The graph for the binomial coefficients resembles a Pascal Triangle, while that for trinomial or multinomial coefficients looks like a Pascal Pyramid, Tetrahedron, or Hyper-Pyramid. Each of the four functions is computed using both its conventional and recursive definitions. Moreover, the recursive structures of the binomial coefficient and the corresponding probability are utilized in an iterative scheme, which is substantially more efficient than the conventional or recursive evaluation. Analogous iterative evaluations of the multinomial coefficient and probability can be constructed similarly. Applications to the reliability evaluation for two-valued and multi-valued k-out-of-n systems are also pointed out.
Keywords: Binomial, multinomial, recursion, boundary condition, signal flow graph, iteration, k-out-of-n.