Communications of the
Korean Mathematical Society CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024
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Commun. Korean Math. Soc. 2002; 17(2): 363-370

Printed June 1, 2002

Copyright © The Korean Mathematical Society.

An asymptotic formula for $\exp(\frac{x}{1-x})$

Junho Song, Changwoo Lee

University of Seoul, University of Seoul

Abstract

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We show that $G(x)=e^{x/(1-x)}-1$ is the exponential generating function for the labeled digraphs whose weak components are transitive tournaments and derive both a recursive formula and an explicit formula for the number of them on $n$ vertices. Moreover, we investigate the asymptotic behavior for the coefficients of $G(x)$ using Hayman's method.

Keywords: transitive tournament, exponential generating function, recursive formula, asymptotics, admissible, Hayman's method

MSC numbers: 05C30, 05C20

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