Cycle decompositions and train tracks

We prove that the disjoint cycle decomposition of the permutation $(1 \, 2 \cdots n_1)^{k_1} (1 \, 2 \cdots n_2)^{k_2} \cdots (1 \, 2 \cdots n_r)^{k_r}$ consists of cycles of at most $r$ distinct lengths. The proof relies on the geometry and topology of simple closed curves and train tracks on a closed surface of genus $r$.