This material has been published in J. Combin. Theory Ser. A 105 (2004), 291-334, the only definitive repository of the content that has been certified and accepted after peer review. Copyright and all rights therein are retained by Elsevier B.V. This material may not be copied or reposted without explicit permission.

Christian Krattenthaler and Thomas W. Müller

Equations in finite semigroups: Explicit enumeration and asymptotics of solution numbers

(39 pages)

Abstract. We study the number of solutions of the general semigroup equation in one variable, Xa=Xb, as well as of the system of equations X2=X, Y2=Y, XY=YX in H\wr Tn, the wreath product of an arbitrary finite group H with the full transformation semigroup Tn on n letters. For these solution numbers, we provide explicit exact formulae, as well as asymptotic estimates. Our results concerning the first mentioned problem generalize earlier results by Harris and Schoenfeld (J. Combin. Theory Ser. A 3 (1967), 122-135) on the number of idempotents in Tn, and a partial result of Dress and the second author (Adv. in Math. 129 (1997), 188-221). Among the asymptotic tools employed are Hayman's method for the estimation of coefficients of analytic functions and the Poisson summation formula.

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