For example, I could think of the notion of an ordered list of linear 'holes' of type α in a datatype T (where T mentions α — perhaps T is the type of binary trees with α at the leaves or something like that). There might be no holes, or one hole, or two holes, or three holes, etc., so the differential operator I want is

1 + D

_{α}+ D

_{α}

^{2}+ D

_{α}

^{3}+ ...

or in other words

(1 / (1 - D

_{α}))

which has the same shape as the algebraic representation of lists as a data structure in the first place.

Of course this guy seems to satisfy mysterious laws like

D

_{α}(1 / (1 - D

_{α}))T + T = (1 / (1 - D

_{α}))T

which makes perfect sense in terms of the algebra of real numbers if you squint and forget that T and D

_{α}are of kind (type -> type) and (type -> type) -> (type -> type) respectively.

I could also do unordered lists of holes by dividing by the number of permutations and get

1 + D

_{α}+ D

_{α}

^{2}/2! + D

_{α}

^{3}/3! + ...

or in other words

exp(D

_{α})

I would definitely conjecture that

(exp(D

_{α}))T = [α+1/α]T

(that is to say, a piece of data of type T with some number of holes where the order doesn't matter is equivalent to replacing the argument α to T with the type α + 1) but I have no clue how I would prove it. It also mysteriously works out in terms of power series of real numbers, though:

f + f' + f''/2! + f'''/3! + ... = f(x+1)

by an easy application of the binomial theorem.