Some asymptotic methods in combinatorics
- 1 November 1979
- journal article
- research article
- Published by Cambridge University Press (CUP) in Journal of the Australian Mathematical Society
- Vol. 28 (4) , 452-460
- https://doi.org/10.1017/s1446788700012593
Abstract
Let 〈fn≧0 be nonnegative real numbers with generating function f(x) = Σfnxn. Assume f(x) has the following properties: it has a finite nonzero radius of convergence x0 with its only singularity on the circle of convergence at x = x0 and f(x0) converges to y0; y = f(x) satisfies an analytic identity F(x, y) = 0 near (x0, y0); Fy(l) (x0, y0)= 0, 0 ≦ i < k and Fy(k) (x0, y0) ≠ 0. There are constants γ, a positive rational, and c such that fn~cx0−n n−(1 +ggr;). Furthermore, we show (i) in all cases how to determine γ and c from f(x) and (ii) in certain cases how to determine them from F(x, y).Keywords
This publication has 3 references indexed in Scilit:
- Twenty-step algorithm for determining the asymptotic number of trees of various specesJournal of the Australian Mathematical Society, 1975
- Asymptotic Methods in EnumerationSIAM Review, 1974
- Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische VerbindungenActa Mathematica, 1937