If $n$ is even, then the [number of permutations of $n$ in which all cycles have odd length](http://oeis.org/A000246) equals the [number of permutations of $n$ in which all cycles have even length](http://oeis.org/A001818). This fact is easily proved, for example using exponential generating functions.



> 
> Is there a simple bijective proof?
> 
> 
> 


Remark: The June 2009 edition of [Ponder This](http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/solutions/June2009.html) sketches a proof by Eytan Sayag but I have not succeeded in understanding it. Perhaps all that is needed is for someone to explain that proof to me in more detail.



Permutations with all cycles odd length and permutations with all cycles even length
Yes, and the original paper is the following. 


MR1774748 Reviewed Bóna, Miklós; McLennan, Andrew; White, Dennis Permutations with roots. Random Structures Algorithms 17 (2000), no. 2, 157–167.


