Idempotent endmorphisms

How many of the endmorphisms of a finite set are idempotent? (Lawvere & Schanuel)

If they’re idempotent, they’re projections. The projections can be onto a nonempty subset, which for sizes [1 .. n], number  [choose n k |k choose n 2,…, choose n n = n, (n-1)n, …, 1

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