This material has been published in
J. Combin. Theory Ser. A
105 (2004), 291-334,
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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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