  In this paper, a new expression for the partition function of the generalized
Penner model given by Goulden, Harer and Jackson is derived. The Penner and the
orthogonal Penner partition functions are special cases of this formula. The
parametrized Euler characteristic $\xi^s_g(\gamma)$ deduced from our expression
of the partition function is shown to exhibit a contribution from the orbifold
Euler characteristic of the moduli space of Riemann surfaces of genus $g$, with
$s$ punctures, for all parameters $\gamma$ and $g$ odd. The other contributions
for $g$ even are linear combinations of the Bernoulli polynomials at rational
arguments. It turns out that the free energy coefficients of the generalized
Penner model in the continuum limit, are identical to those coefficients in the
large $N$ expansion of the Gaussian $\beta$-ensemble. Moreover, the duality
enjoyed by the generalized Penner model, is also the duality symmetry of the
Gaussian $\beta$-ensemble. Finally, a shift in the 't Hooft coupling constant
required by the refined topological string, would leave the Gaussian
$\beta$-ensemble duality intact. This duality is identified with the remarkable
duality of the $c=1$ string at radius $R=\beta$.
