  We prove some combinatorial results related to a formula on Hodge integrals
conjectured by Mari\~no and Vafa. These results play important roles in the
proof and applications of this formula by the author jointly with Chiu-Chu
Melissa Liu and Kefeng Liu. We also compare with some related results on
Hurwitz numbers and obtain some closed expressions for the generating series of
Hurwitz numbers and the related Hodge integrals.
