Let $c(n,k)$ be the number of permutations of $[n]$ with $k$ cycles. I am looking for a proof of the following.
- $c(n,k)=(n-1)c(n-1,k)+c(n-1,k-1)$ for $n,k \geq 1$ and $c(0,0)=1$
- The number of permutations $\pi \in S_n$ with cycle type $(c_1,\dots,c_n)$ is $\frac{n!}{1^{c_1}c_1!2^{c_2}c_2!\dots n^{c_n}c_n!}$ where $c_i=c_i(\pi)$ is the number of cycles of $\pi$ of length $i$.
- $\sum\limits_{k=0}^{n} c(n,k)t^k=t(t+1)(t+2)\dots(t+n-1)$.
- $\sum\limits_{k=0}^{n} c(n,k) \frac{x^n}{n!}=\frac{1}{k!}\left(\log(\frac{1}{1-x}) \right) ^k$
- I looking for a bijection between the set of permutations with $k$ cycles and the set of permutations with $k$ left-right minima.
Thanks for your help!