Combinatorics, equality, $n$-permutations with $k$ cycles

Asked Sep 15 '14 at 05:43

Active Sep 15 '14 at 05:43

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Let $b_r(n,k)$ denote the number of $n$-permuations with $k$ cycles in which all numbers: $1,...,r$ are in one cycle.

Prove that for $n \ge r$, $\sum _{k=1} ^n b_r(n,k) x^k = (r-1)! \frac{x^{\overline{n}}}{(x+1)^{\overline{r-1}}}$.

Here $x^{\overline{n}} = x(x+1)...(x+n-1)$

I've been thinking of using induction to prove it. But I don't know which variable to choose - $r$ or $n$ or somehow both?

Could you help me with proving this formula?

Thank you!

asked Sep 15 '14 at 05:43

Bilbo

2

If you're not sure, try both! Here induction on $n$ works. – Yuval Filmus Sep 15 '14 at 13:45
There is a solution at this MSE link which is admittedly not the simplest / best but may perhaps prove useful reading. – Marko Riedel Sep 15 '14 at 20:24

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