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Asymptotic Multivariate Enumeration in Combinatorial Probability | |
The following sources are recommended by a professor whose research specialty is multivariate enumeration. |
-- SHORT DESCRIPTION: Given an array of numbers, indexed by d-tuples of non-negative integers, how may one determine asymptotic formulae for these numbers as the indices go to infinity in all possible ways? In one dimension, many techniques are available via complex variable methods. For multivariate arrays, much less is known, but multivariate contour integration yields some answers.
· Bender, E., and Richmond, L.B. (1983). Central and local limit theorems applied to asymptotic enumeration II: multivariate generating functions. J. Combin. Theory Ser. A, 34, 255-265.
· Bleistein, N., and Handelsman, R. (1986). Asymptotic expansions of integrals, second edition. Dover.
· Flajolet, P., and Odlyzko, A. (1990). Singularity analysis of generating functions. SIAM J. Disc. Math, 3, 216-240.
· Gao, Z., and Richmond, L.B. (1992). Central and local limit theorems applied to asymptotic enumeration IV: ultivariate generating functions. J. Comput. Appl. Math., 41, 177-186.
· Odlyzko, A. (1995). Asymptotic enumeration methods. In: Handbook of combinatorics, vol. II, Graham, Grötschel, and Lovasz, editors. Elsevier.
· Pemantle, R., and Wilson, M. (2001). Asymptotics of multivariate sequences, part I: smooth points of the singular variety. J. Combin. Theory Ser. A, 97, 129-161. http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WHS-458NC34-N&_coverDate=01%2F31%2F2002&_alid=160436274&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=6858&_sort=d&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=7476333e77e6c991e22a8dc4d10d6767
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