Hodge integrals, Hurwitz numbers, and Symmetric Groups

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.