I have been trying to solve this problem in the book A Walkthrough Combinatorics. However, I am struggling. There is a duplicate but it has never been answered.
We have n cards. We want to split them into an even number of non-empty subsets, form a line within each subset, and then arrange the subsets in a line. In how many different ways can we do this?
I think I should use the composition rule of generating functions. Taking $A_n(x)$ (forming a line within each subset, which is an even number) = $\sum_{n\geq0} 2^{2n-1} x^{2n}$, and $B_n(x)$ (arranging subsets in a line) = $\sum_{n\geq0} n! x^{2n}$
I think $a_n$ should be $2^{n-1}$ since this is how many lines we can form from n elements. Is this true? I feel like there are some huge parts missing.