  Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$
an ordered domain. We consider the algorithmic problem of computing the
generalized Euler-Poincar\'e characteristic of real algebraic as well as
semi-algebraic subsets of $\mathrm{R}^k$, which are defined by symmetric
polynomials with coefficients in $\mathrm{D}$. We give algorithms for computing
the generalized Euler-Poincar\'e characteristic of such sets, whose
complexities measured by the number the number of arithmetic operations in
$\mathrm{D}$, are polynomially bounded in terms of $k$ and the number of
polynomials in the input, assuming that the degrees of the input polynomials
are bounded by a constant. This is in contrast to the best complexity of the
known algorithms for the same problems in the non-symmetric situation, which
are singly exponential. This singly exponential complexity for the latter
problem is unlikely to be improved because of hardness result
($\#\mathbf{P}$-hardness) coming from discrete complexity theory.
