  A new algebraic formula for the numbers of faces of nestohedra is obtained.
The enumerator function $F(P_B)$ of positive lattice points in interiors of
maximal cones of the normal fan of the nestohedron $P_B$ associated to a
building set $B$ is described as a morphism from the certain combinatorial Hopf
algebra of building sets to quasisymmetric functions. We define the $q$-analog
$F_q(P_B)$ and derive its determining recurrence relations. The $f$-polynomial
of the nestohedron $P_B$ appears as the principal specialization of the
quasisymmetric function $F_q(P_B)$.
