It’s counting time

What is the number of idempotent functions f:\{1,\dots,n\}\to\{1,\dots,n\}?

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It’s counting time

Points on the unit circle

This problem is from Stanley’s “Enumerative Combinatorics Vol. I”, more precisely problem 12 from Chapter 1 (sadly, no solution is provided there.)

Consider n points arranged in the unit circumference and draw all \binom n2 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 2n points in the circle, and label n of them with a 1, and the remaining n with a -1. 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?

 

Points on the unit circle