Yesterday, as a puzzle, I asked what the generating function
counts. Today I’ll give some hints; unfortunately I did not have enough time to write up a satisfying solution. As commenter Zahlen points out, it’s better to think of , not as an ordinary generating function, but as an exponential generating function. That makes it the exponential generating function of the sequence of squared factorials.
The reason we’d like to do this, although Zahlen didn’t make this explicit, is that this maneuver opens up the possibility of appealing to the exponential formula. Loosely speaking, the exponential formula can be interpreted as saying that if some exponential generating function counts objects that have a decomposition into “connected components,” then counts connected objects. For example, when (the exponential generating function of the factorials), the logarithm is
and this can be interpreted as reflecting the cycle decomposition of a permutation.
So: what is the relevant “connected components” decomposition here?