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Dr Sardonicus · 2021-10-30, 15:32

If n > 1 is an integer, a permutation $\sigma\;\in\;S_{n}$ - the group of permutations of the set of integers from 1 to n - is called a derangement if it has no fixed points; that is, $\sigma(k)\;\ne\;k$ for k = 1 to n.

The number of derangements in S_n may be determined by "inclusion-exclusion." The proportion of elements in S_n which are derangements is

$\sum_{i=0}^{n}\frac{(-1)^{i}}{i!}$

which is a partial sum of the Taylor series for 1/e.

2021-10-30, 15:32	#4
Dr Sardonicus Feb 2017 Nowhere 2×11×263 Posts	If n > 1 is an integer, a permutation $\sigma\;\in\;S_{n}$ - the group of permutations of the set of integers from 1 to n - is called a derangement if it has no fixed points; that is, $\sigma(k)\;\ne\;k$ for k = 1 to n. The number of derangements in S_n may be determined by "inclusion-exclusion." The proportion of elements in S_n which are derangements is $\sum_{i=0}^{n}\frac{(-1)^{i}}{i!}$ which is a partial sum of the Taylor series for 1/e. Last fiddled with by Dr Sardonicus on 2021-10-30 at 15:33 Reason: xignif topsy