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\iteman{io-port 05701972}
\itemau{Kim, Jang Soo}
\itemti{A note on the total number of cycles of even and odd permutations.}
\itemso{Discrete Math. 310, No. 8, 1398-1400 (2010).}
\itemab
Summary: We prove bijectively that the total number of cycles of all even permutations of $[n]=\{1,2,\dots ,n\}$ 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 {\it Mikl\'os B\'ona} [Combinatorics of permutations, Discrete Mathematics and its Applications, Boca Raton, FK: Chapman \& Hall / CRC (2004; Zbl 1052.05001)]. 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.
\itemrv{~}
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\itemut{cycles of permutations; cycles of even permutations; cycles of odd permutations; sign-reversing involution}
\itemli{doi:10.1016/j.disc.2009.12.018}
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