  We study the number of solutions of the general semigroup equation in one
variable, $X^\al=X^\be$, as well as of the system of equations $X^2=X, Y^2=Y,
XY=YX$ in $H\wr T_n$, the wreath product of an arbitrary finite group $H$ with
the full transformation semigroup $T_n$ 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 $T_n$, 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.
