A note on the total number of cycles of even and odd permutations

We prove bijectively that the total number of cycles of all even permutations of [n] = {1, 2, ..., n} and the total number of cycles of all odd permutations of [n] differ by (- 1)ⁿ (n - 2) !, which was stated as an open problem by Miklós Bóna. We also prove bijectively the following more general identity: underover(∑, i = 1, n) c (n, i) {dot operator} i {dot operator} (- k)^{i - 1} = (- 1)^k k ! (n - k - 1) !, where c (n, i) denotes the number of permutations of [n] with i cycles.