Let $\pi_n$ be the poset of all set partitions of $\{1,...,n\}$ ordered by refinement, $\sigma = \{B_1,...,B_k\}$ be a set partition with blocks $B_i$, and $\max(B_i)$ be the maximum value in the block $B_i$. I'm trying to prove the following:

$$\Sigma_{\sigma\in\pi_{n-1}}\Pi_{B_i\in\sigma}(-1)^{|B_i|-1}(|B_i|-1)!(n+\max(B_i))=(2n-1)!!$$

I've already coded this in Mathematica to check that it's true until n=11 (after which the numbers become too large):

```
<<Combinatorica`
f[x_]:=(-1)^(x-1)*(x-1) !
Total[Times @@@ Map[f, Map[Length, SetPartitions[n-1], {2}], {2}]*Times @@@ (Map[Max, SetPartitions[n-1], {2}] + n) ] == Factorial2[2*n-1]
```

Any thoughts would be greatly appreciated.

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