Efficient numerical evaluation of weak restricted compositions
We propose an algorithm to calculate the number of weak compositions, wherein each part is restricted to a different range of integers. This algorithm performs different orders of approximation up to the exact solution by using the Inclusion-Exclusion Principle. The great advantage of it with respect to the classical generating function technique is that the calculation is exponentially faster as the size of the numbers involved increases.
Brualdi RA. Introductory Combinatorics. Pearson Education India; 1977.
Comtet L. Advanced Combinatorics: The art of infinite expansions. Springer Science & Business Media, 2012.
De Moivre A. The doctrine of chances: or, A method of calculating the probabilities of events in play, vol. 200. Chelsea Publishing Company Incorporated, 1756.
Eger S. Restricted weighted integer compositions and extended binomial coefficients. J Integer Seq 2013;16(13.1.3):1-25.
Feller W. An introduction to probability theory and its applications, vol. 2. John Wiley & Sons, 2008.
Heubach S, Mansour T. Compositions of n parts in a set. Congressus Numerantium 2004;168:127-140.
Heubach S, Mansour T. Combinatorics of compositions and words. CRC Press, 2009.
Jakli? G, Vitrih V, Žagar E. Closed form formula for the number of restricted compositions. Bulletin of the Australian Mathematical Society 2010;81(2):289-297. doi:10.1017/S0004972709000902