Theta divisors and permutohedra

We establish an intriguing relation of the smooth theta divisor ${\mathrm{\Theta }}^n$ with permutohedron ${\mathrm{\Pi }}^n$ and the corresponding toric variety $X_{\mathrm{\Pi }}^n\mathrm{.}$ In particular, we show that the generalised Todd genus of the theta divisor ${\mathrm{\Theta }}^n$ coincides with $h$-polynomial of permutohedron ${\mathrm{\Pi }}^n$ and thus is different from the same genus of $X_{\mathrm{\Pi }}^n$ only by the sign $(-1{)}^n\mathrm{.}$ As an application we find all the Hodge numbers of the theta divisors in terms of the Eulerian numbers. We reveal also interesting numerical relations between theta-divisors and Tomei manifolds from the theory of the integrable Toda lattice.