Hi, I have a question that requires the use of generating functions in combinatorics;
Let t(n) be the number of ways to arrange n books on two bookshelves so that each shelf receives at least one book. Find a closed formula for tn.
Thanks
Hi, I have a question that requires the use of generating functions in combinatorics;
Let t(n) be the number of ways to arrange n books on two bookshelves so that each shelf receives at least one book. Find a closed formula for tn.
Thanks
I'm assuming the order of the books on each shelf is important.
If we have n books on a single shelf, there are n! ways to arrange the books. If we require n > 0, the Exponential Generating Function (EGF) of n! is
For books on two shelves with at least one book on each shelf, the EGF is then
The coefficient of in the above series is
.