Lesson 7

It can be calculated directly from the definition: for each of the \(n!\) permutations, write down how many fixed points it has, then take their average. As \(n\) increases, this becomes more difficult both on paper and on a computer.

from itertools import permutations
from statistics import mean

n = 8
totals = []
for p in permutations(range(n)):
    fixed = [i for i in range(n) if p[i] == i]
    totals.append(len(fixed))

print(mean(totals))

1