June 7, 2026
Bell numbers, big feelings
Partitions over Permutations
A math blog tried to tame a wild formula, and the comments instantly went full nerd chaos
TLDR: A math blogger found that a clever shortcut breaks badly in one direction and leads to a monster function tied to counting ways to group labeled things. In the comments, readers immediately turned it into a showy explain-off, with the biggest mood being: nice post, but let’s make it even nerdier.
A seemingly innocent math post about a quirky shortcut for the bell-curve formula somehow turned into a full-blown comment-section flex. The author noticed that a clever cosine-based approximation behaves nicely in one direction, then goes completely off the rails in another, exploding far faster than expected. That sent readers down a rabbit hole about the dizzying growth of the “double exponential” function — which, in plain English, is a number monster that gets huge ridiculously fast. The real twist: its coefficients are tied to Bell numbers, which count how many ways you can group labeled items, and that invited the classic internet math energy of “cool fact” immediately followed by “well, actually.”
The strongest reaction came from readers eager to out-explain the post in the comments. One commenter swooped in with the symbolic method, basically saying this whole thing has a tidy combinatorics explanation if you speak the secret language of counting tricks. That set the tone: less “wow, neat” and more “allow me to make this even more abstract.” The mini-drama here is deliciously nerdy — the post tried to make a surprising pattern feel intuitive, while the comments raced to formalize it, decode it, and gently show off. Even the footnote about labeled versus unlabeled sets feels like preemptive damage control against the inevitable correction squad. The vibe was pure math internet: half awe, half one-upmanship, and 100% delighted by gigantic numbers behaving badly.
Key Points
- •The article compares `exp(-z²)` with `(1 + cos(sin(z) + z))/2` and says the approximation fails along the imaginary axis, where the latter behaves like `exp(exp(y))`.
- •It states that the coefficient of `x^n` in the power series for `exp(exp(y))` is `e Bn / n!`.
- •The article identifies `Bn` as the nth Bell number, counting partitions of a labeled set of `n` items, while `n!` counts permutations of those items.
- •It says Bell numbers grow almost as quickly as factorials, which makes the double exponential series converge slowly.
- •The article provides asymptotic formulas for `log(Bn / n!)`, including one expressed with the Lambert W function, and notes that Bell numbers differ from partition numbers for unlabeled sets.