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: ∑ i = 1 n c ( n , i ) ⋅ i ⋅ ( − k ) i − 1 = ( − 1 ) k k ! ( n − k − 1 ) ! , where c ( n , i ) denotes the number of permutations of [ n ] with i cycles.