[re-sent for mistyped header--apologies if duplicated] wouter> the reverse operation is amazing:
just keep splitting any even number into halves until no evens remain!
And the inverse of *that* is: Keep pairwise adding repeated parts until no pairs remain. There are *two* similar bijections of not-all-odd with not-all-different. Note that the original can be phrased: parts differ by at least 1 <-> parts = +-1 mod 4. It's also true that parts differ by at least 2 <-> parts = +-1 mod 5. Good luck with *that* bijection! Now that the cat is out of the bag, I hope Andrew Granville won't mind me publicly stating that his modular reciprocal theorem is news to me and worthy of Knuth's attention as a candidate exercise for Volume II. --Bill Gosper PS: I just rediscovered the denesting of sqrt(4^(1/5)-3^(1/5)). I probably mentioned it the first time. If not, it's a decent puzzle.