In this talk, we introduce an explicit description for the rational Betti numbers of the real toric manifolds associated to a chordal building set $\mathcal{B}$ in terms of alternating $\mathcal{B}$-permutations.

For a chordal graph with a perfect elimination ordering, we give a combinatorial interpretation of the graph $a$-numbers.

In addition, we provide detailed computations for some interesting chordal nestohedra; permutohedra, associahedra, stellohedra, Stanley-Pitman polytopes, Hochschild polytopes.

This is joint work with Suyoung Choi.