2.2: Forbidden Position Permutations
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- 7146
Suppose we shuffle a deck of cards; what is the probability that no card is in its original location? More generally, how many permutations of $[n]=\{1,2,3,\ldots,n\}$ have none of the integers in their "correct'' locations? That is, 1 is not first, 2 is not second, and so on. Such a permutation is called a derangement of \([n]\).
Let \(S\) be the set of all permutations of \([n]\) and \(A_i\) be the permutations of \([n]\) in which \(i\) is in the correct place. Then we want to know $|\bigcap_{i=1}^n A_i^c|$.
For any \(i\), \(|A_i|=(n-1)!\): once \(i\) is fixed in position \(i\), the remaining \(n-1\) integers can be placed in any locations.
What about $|A_i\cap A_j|$? If both \(i\) and $j$ are in the correct position, the remaining \(n-2\) integers can be placed anywhere, so $|A_i\cap A_j|=(n-2)!$.
In the same way, we see that $|A_{i_1}\cap A_{i_2}\cap\cdots\cap A_{i_k}|=(n-k)!$. Thus, by the inclusion-exclusion formula, in the form of equation 2.1.1,
\[\eqalign{ |\bigcap_{i=1}^n A_i^c|&=|S|+\sum_{k=1}^n (-1)^k{n\choose k}(n-k)!\cr &=n!+\sum_{k=1}^n (-1)^k{n!\over k!(n-k)!}(n-k)!\cr &=n!+\sum_{k=1}^n (-1)^k{n!\over k!}\cr &=n!+n!\sum_{k=1}^n (-1)^k{1\over k!}\cr &=n!\,\Bigl(1+\sum_{k=1}^n (-1)^k{1\over k!}\Bigr)\cr &=n!\,\sum_{k=0}^n (-1)^k{1\over k!}.\cr }\]
The last sum should look familiar: $$e^x=\sum_{k=0}^\infty {1\over k!}x^k.$$ Substituting $x=-1$ gives $$e^{-1} = \sum_{k=0}^\infty {1\over k!}(-1)^k.$$ The probability of getting a derangement by chance is then $${1\over n!}n!\,\sum_{k=0}^n (-1)^k{1\over k!} =\sum_{k=0}^n (-1)^k{1\over k!},$$ and when $n$ is bigger than 6, this is quite close to $$e^{-1} \approx 0.3678.$$ So in the case of a deck of cards, the probability of a derangement is about 37%.
Let $D_n=n!\,\sum_{k=0}^n (-1)^k{1\over k!}$. These derangement numbers have some interesting properties. The derangements of \([n]\) may be produced as follows: For each $i\in\{2,3,\ldots,n\}$, put \(i\) in position 1 and 1 in position \(i\). Then permute the numbers $\{2,3,\ldots,i-1,i+1,\ldots n\}$ in all possible ways so that none of these $n-2$ numbers is in the correct place. There are $D_{n-2}$ ways to do this. Then, keeping 1 in position \(i\), derange the numbers $\{i,2,3,\ldots,i-1,i+1,\ldots n\}$, with the "correct'' position of \(i\) now considered to be position 1. There are $D_{n-1}$ ways to do this. Thus, $D_n=(n-1)(D_{n-1}+D_{n-2})$.
We explore this recurrence relation a bit: $$\eqalignno{ D_n&=nD_{n-1}-D_{n-1}+(n-1)D_{n-2}&(*)\cr &=nD_{n-1}-(n-2)(D_{n-2}+D_{n-3})+(n-1)D_{n-2}\cr &=nD_{n-1}-(n-2)D_{n-2}-(n-2)D_{n-3}+(n-1)D_{n-2}\cr &=nD_{n-1}+D_{n-2}-(n-2)D_{n-3}&(*)\cr &=nD_{n-1}+(n-3)(D_{n-3}+D_{n-4})-(n-2)D_{n-3}\cr &=nD_{n-1}+(n-3)D_{n-3}+(n-3)D_{n-4}-(n-2)D_{n-3}\cr &=nD_{n-1}-D_{n-3}+(n-3)D_{n-4}&(*)\cr &=nD_{n-1}-(n-4)(D_{n-4}+D_{n-5})+(n-3)D_{n-4}\cr &=nD_{n-1}-(n-4)D_{n-4}-(n-4)D_{n-5}+(n-3)D_{n-4}\cr &=nD_{n-1}+D_{n-4}-(n-4)D_{n-5}.&(*)\cr }$$ It appears from the starred lines that the pattern here is that $$D_n=nD_{n-1}+(-1)^kD_{n-k}+(-1)^{k+1}(n-k)D_{n-k-1}.$$ If this continues, we should get to $$D_n=nD_{n-1}+(-1)^{n-2}D_{2}+(-1)^{n-1}(2)D_{1}.$$ Since $D_2=1$ and $D_1=0$, this would give $$D_n=nD_{n-1}+(-1)^n,$$ since $\displaystyle (-1)^n=(-1)^{n-2}$. Indeed this is true, and can be proved by induction. This gives a somewhat simpler recurrence relation, making it quite easy to compute $D_n$.
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There are many similar problems.
Example \(\PageIndex{1}\):
How many permutations of \([n]\) contain no instance of \(i\) followed by $i+1$?
By a similar use of the inclusion-exclusion formula, it turns out that this is $$Q_n=n!\,\sum_{k=0}^{n-1} (-1)^k{1\over k!}+ (n-1)!\,\sum_{k=1}^{n-1} (-1)^{k-1} {1\over (k-1)!}. $$ Note that the limits on the two sums are not identical.