Q: We need to divide $n$ cards into groups that are even(number of groups are $2k$, but number of the elements of the groups can be odd numbers). Then we need to list each group in a line after that we need to list the groups in a line. How many ways to get that?
I have an answer but it does not seems right. I tried to achieve this question in simple cases. Assume we will have $2$ groups. We will have $a_n$ for first group and $b_m$ for second group. Since we need to list the elements in a line we'll have:
$a_n=n!$ and $b_m=m!$ (since we need to line them) and then I'll look the exponential generating functions:
$A(x)=\sum_{n\geq0}n!\frac{x^n}{n!} = \frac{1}{1-x}$ and $B(x)=\sum_{k\geq0}k!\frac{x^k}{k!} = \frac{1}{1-x}$
After that point, I multiplied these two functions and then I multiply with $2!$ which is $\frac{2!}{(1-x)^{2}}$. After that I tried to generalize it and I obtained:
$\sum_{n\geq1}\frac{2n!}{(1-x)^{2n}}$.
Can you please help me about that question?
HINT: In effect you just have a permutation of the $n$ cards into which you’ve inserted $2k-1$ dividers to break it into $2k$ segments. There are $n-1$ slots between cards, and you can insert at most one divider into any of those slots.