What is the number of idempotent functions ?
This problem is from Stanley’s “Enumerative Combinatorics Vol. I”, more precisely problem 12 from Chapter 1 (sadly, no solution is provided there.)
Consider points arranged in the unit circumference and draw all chords connecting two of the points. Assume that the points are laid out in such a way no three chords intersect in a single point (that is, there are no triple intersections). Into how many regions will the interior of the circle by divided? Stanley says: “Try to give an elegant proof avoiding induction, finite differences, generating functions, summations, and the like.” One can do such a thing.
There’s another problem that involves placing points in the circle. Place points in the circle, and label of them with a , and the remaining with a . Prove we can trace the circle, starting from on of this points, in such a way that the partial sums of the values of the points is always positive, and this can be done both clockwise and anticlockwise. Can you generalize this to a continuous (as opposed to discrete) setting?
Consider the infinite product , and write this as . Relate to the binary expression of . The sequence starts off as: .