Using the celebrated Witten–Kontsevich theorem, we prove a recursive formula of the $n$-point functions for intersection numbers on moduli spaces of curves. It has been used to prove the Faber intersection number conjecture and motivated us to find some conjectural vanishing identities for Gromov–Witten invariants. The latter has been proved recently by Liu and Pandharipande. We also give a combinatorial interpretation of $n$-point functions in terms of summation over binary trees.
"The $n$-point functions for intersection numbers on moduli spaces of curves." Adv. Theor. Math. Phys. 15 (5) 1201 - 1236, October 2011.