Descent set distribution for permutations with cycles of only odd or only even lengths

It is known that the number of permutations in the symmetric group S 2 n with cycles of odd lengths only is equal to the number of permutations with cycles of even lengths only. We prove a refinement of this equality, involving descent sets: the number of permutations in S 2 n with a prescribed descent set and all cycles of odd lengths is equal to the number of permutations with the complementary descent set and all cycles of even lengths. There is also a variant for S 2 n + 1 . The proof uses generating functions for character values and applies a new identity on higher Lie characters.