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A derangement of a list of $n$ distinct entries is a permutation of that list such that no corresponding entries match. It is well-known that the number of such derangements is the nearest integer to $n!/e$ where $e$ is the base of natural logarithms.

Let's say that a permutation of a list is a $\mu$-derangement if both it and its reverse ordering are derangements. Equivalently, iff the permutation is a derangement both of the original list and of the reverse of the original list.

How many $\mu$-derangements of the list $[1,2,..,n]$ are there? Is there an exact formula? A good approximation or bound?

There are no $\mu$-derangements for $n \lt 4$, except in the trivial case of an empty list. I got counts for $n \le 10$ below by enumerating possibilities with a bit of Prolog code:

 n | # of µ-derangements 
---+----------------------
 4 |           4
 5 |          16
 6 |          80
 7 |         672
 8 |        4752
 9 |       48768
10 |      440192

The OEIS has this sequence as A003471, with a recurrence relation that suggests some separation into even and odd terms might simplify things.

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  • I think you mean $n!/e$- is it really the nearest integer to that? I know that the probability that a permutation is a derangement is close to $1/e.$ – Geoff Robinson Dec 27 '11 at 11:21
  • @GeoffRobinson: Just so, Geoff. Fixed! – hardmath Dec 27 '11 at 11:26
  • @GeoffRobinson: Peter Cameron's Lectures on Derangements presents three proofs that the number of derangements is the closest integer to $n!/e$, the first being the most elementary. – hardmath Dec 27 '11 at 11:57
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    Empirically, both sets of values seem to be asymptotic to $n!/e^2$. The dominant error term appears to be $1/n$ for even values of $n$ and $1/(2n^2)$ for odd values. Probably an asymptotic expansion is possible based on the recurrence relation? – mjqxxxx Dec 27 '11 at 13:02
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    Note that the very rough approximation $(1- \frac{2}{n})^n$ gives the right answer asymptotically for the probability that a permutation is this kind of derangement, even though it (unjustifiably) treats the choices as independent. The heuristic argument is that for a given permutation $\sigma,$ there are two values of $\sigma(i)$ which are not permissible. – Geoff Robinson Dec 27 '11 at 14:58
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    I proposed an edit to the OEIS sequence adding the integral formula. – joriki Dec 27 '11 at 15:19
  • Thanks for that (proposed) addition, @joriki! – J. M. ain't a mathematician Dec 28 '11 at 02:43

1 Answers1

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For $n=2k$, this is equivalent to What's the General Expression For Probability of a Failed Gift Exchange Draw, with a slot and its reverse partner forming a couple – the answer for this case is

$$\int_0^\infty\left(x^2-4x+2\right)^k\mathrm e^{-x}\mathrm dx\;,$$

and I derived in the answer to the other question that this goes as $n!/\mathrm e^2$ for large $n$.

For $n=2k+1$, there's a single slot without a partner which leads to a factor $-L_1(x)=x-1$, so the answer for this case is

$$\int_0^\infty\left(x^2-4x+2\right)^k(x-1)\mathrm e^{-x}\mathrm dx\;.$$

The asymptotic analysis remains essentially unchanged, so this also goes as $n!/\mathrm e^2$ for large $n$.

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  • Danke shön, joriki! The application that motivated me here was for an $n = 2k$ even term case, trying to optimize the search strategy there. Your answer nicely sheds light on why the even and odd term cases differ by a little. Since $\int_0^\infty x^k e^{-x} dx = k!$, one can in principle expand your integral formula into a summation involving factorials. – hardmath Dec 27 '11 at 19:26
  • @hardmath: It does seem so; expand the polynomial power with the multinomial formula, and then replace the integrals with factorials... if there was a recurrence relation for these, I wouldn't be surprised. – J. M. ain't a mathematician Dec 28 '11 at 02:45