Permutations With No Identity Elements

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Number of permutations where n ≠ position n

There are $N!$ permutations of the set $\{1,2,\ldots,N\}$

How many of them have zero identity elements?

An identity element is an element that has a value equal to its position. ie When for some $i$, the ith element equals $i$.

For example, $(2,3,4,1)$ has no identity elements, whereas $(2,1,3,4)$ has two identity elements.

These are called derangements. This math.SE question, and this one, cover the answer to your question. — Zev Chonoles, Jun 16 '12 at 16:08
Elements mapped to themselves by a permutation are called fixed points, not "identity elements". — hardmath, Jun 16 '12 at 17:02
http://math.stackexchange.com/questions/159158/counting-derangements-explanation-of-wp-description — Andrew Tomazos, Jun 16 '12 at 18:40

score 4 · Accepted Answer · answered Jun 16 '12 at 16:08

4

The usual terminology would be "fixed points" or "1-cycles" instead of "identity elements".

A permutation with no fixed points is called a derangement. The Wikipedia article on derangements gives quite a bit of information about counting them.

answered Jun 16 '12 at 16:08

Jim Belk

49,278

http://math.stackexchange.com/questions/159158/counting-derangements-explanation-of-wp-description – Andrew Tomazos Jun 16 '12 at 18:39

Permutations With No Identity Elements

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