Partitions over permutations

I was thinking more about the cosine approximation to the Gaussian exp(−z²) ≈ (1 + cos(sin(z) + z))/2 that I wrote about last week. The two expressions above are close along the real axis but not along the imaginary axis. If z = iy, the right side grows much faster than the left, behaving like exp(exp(y)). This led to me looking up the power series for the double exponential function exp(exp(y)). This is an interesting series because the coefficient of xn is e Bn / n! where Bn is the nth Bell number, which equals the number of ways to partition a set of n labeled items [1]. And of course n! is the number of ways to permute a set of n labeled items. So the nth coefficient in the power series for exp(exp(y)) is the ratio of the number of partitions to permutations for a set of n labeled things, multiplied by e. The number of ways to partition a set of n things grows quickly as n increases, almost as quickly as the number of permutations, and so the series for the double exponential…