What Is the number of regular graphs of degree 2 on $n$ vertices?
I came up with this question in my graph theory textbook where the answer is given as the number of partitions of $n$ where each part is greater than 3; Wolfram Mathworld gives a formula in terms of the partition function without explanation...Can someone explain?How can we draw a bijection between them?
The key thing to note that leads to partitions is the fact that if your $2$-regular graph is connected, then it must just be a cycle. As such, calculating the number of $2$-regular graphs on $n$ vertices amounts to counting the number of ways of dividing the vertices into groups and then ordering them into a cycle.