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My coauthor Dave Perkinson made this diagram to explain the binomial theorem and I think it's the perfect distillation. Here's our draft book, Discrete Structures, if you want more: kyleormsby.github.io/files/113spr... 🧮

You also might appreciate this use of color for illustrating a fancier binomial theorem (§2.2 of arxiv.org/abs/2309.03181):

What do you mean by the ghost coordinates of a G-set? The orbit decomposition? If R is a Green functor, you can define ideals I_H = fib(R^G → R^H) (If R doesn't have surjective restrictions then I_H is a "quasi-ideal" i.e. a derived thing.) Then you can let (n)_R! := ⊗_{|H| ≤ n} I_H

For G=𝕋, this recovers (the ideal generated by) (n)_q! for [H ↦ π₀ ku^tH ] and (q-1)^n (n)_q! for [H ↦ π₀ ku^hH]. Note that here I_{C_n} = π_{λ^n}, using the cell structure 𝕋/C_n → S⁰ → S^{λ^n} and λ⁰ + … + λ^{n-1} = Ind^{C_n}_e(ℂ); this is related to the slice filtration. Idk if that helps

"Why should norms be related to factorials?" any time you see x^n/n! we should be able to deform that to x^{G/H} -------- (G/H)! whatever that means; the above is one candidate for what the denominator should mean