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

Abstract

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: $$\sum _{i=1}^{n}c(n,i)\cdot i\cdot (-k{)}^{i-1}=(-1{)}^{k}k!(n-k-1)!,$$ where $c(n,i)$ denotes the number of permutations of $[n]$ with $i$ cycles.

arXiv comment4 pages, 2 figures, final version