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I'm looking for the method by which the partial derangement formula $D_{n,k}$ was derived. I can determine the values for small values of N empirically, but how the general case formula arose still alludes me.

Any links/books or an explanation will be appreciated.

The formula is:

$D_{n,k} = {n \choose k}!(n-k)$

Links:

  1. Mathworld
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It would be a good idea to define D_{n,k} and maybe to write down the formula as well. I assume your indexing is the same as en.wikipedia.org/wiki/Rencontres_numbers ? –  Qiaochu Yuan Jan 13 '11 at 1:10
    
Yes thats the link i've been using plus the mathworld one. –  Samonir Jan 13 '11 at 1:20
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You are asking why the number of permutations of an $n$-element set with exactly $k$ fixed points is equal to the number of $k$-element subsets of an $n$-element set times the number of permutations of an $(n-k)$-element set with no fixed points, and your question is answered by Douglas S. Stones at the following link: math.stackexchange.com/questions/14477/… –  Jonas Meyer Jan 13 '11 at 1:22
    
@Jonas: That is just a restatement of the forumla, I want to know how it was derived, not interested in what it means. Why the !n, why n-choose-k? did someone just brute force every meaningful combination of combinatorial and permuation math until something clicked? or was there some thought behind it? So far all I can see is the the sum of D_n,k[0,] = n! –  Samonir Jan 13 '11 at 1:25
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@Seminar: I'm sorry you're not interested in what it means, but Douglas S. Stones gives the reason it is true in the linked answer. Thinking about what it means is actually very helpful in coming up with how it is derived. –  Jonas Meyer Jan 13 '11 at 1:27
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2 Answers

up vote 10 down vote accepted

Here is a very general solution. There is a fundamental formula in combinatorics called the exponential formula, and one statement of it is as follows. Given a finite group $G$ acting on a set $X$, its cycle index polynomial is given by

$$Z_G = \frac{1}{|G|} \sum_{g \in G} z_1^{c_1(g)} z_2^{c_2(g)} ... $$

where $c_i(g)$ is the number of cycles of length $i$ in the action of $g$ on $X$. In particular, the notation $Z_{S_n}$ will denote the cycle index polynomial of $S_n$ acting on an $n$-element set in the usual way; it is a generating function encoding the relative proportions of different cycle types of permutations.

The exponential formula then states that

$$\sum_{n \ge 0} Z_{S_n} t^n = \exp \left( z_1 t + \frac{z_2 t^2}{2} + \frac{z_3 t^3}{3} + ... \right).$$

In my opinion this is one of the most beautiful formulas in mathematics and a major reason I became interested in combinatorics was because I stumbled upon this formula while solving a Putnam problem (which is described in the blog post I linked to above).

How does it apply to this problem? Set $z_2 = z_3 = ... = 1$ and $z_1 = z$. Then the LHS of the exponential formula is a generating function which counts permutations according to how many fixed points ($1$-cycles) they have. In other words,

$$Z_{S_n}(z, 1, 1, ...) = \frac{1}{n!} \sum_{g \in S_n} z^{c_1(g)} = \frac{1}{n!} \sum_{k=0}^n D_{n,k} z^k.$$

The RHS of the exponential formula, on the other hand, is

$$\exp \left( zt + \log \frac{1}{1-t} - t \right) = \frac{e^{-t}}{1 - t} e^{zt}.$$

So we obtain the beautifully concise formula

$$\sum_{n \ge 0} \frac{t^n}{n!} \sum_{k=0}^n D_{n,k} z^k = \frac{e^{-t}}{1 - t} e^{zt}.$$

The coefficients of $\frac{e^{-t}}{1 - t}$ are obtained by setting $z = 0$; they give the usual derangement numbers, e.g. the number of permutations of $n$ elements with no fixed points, and this can also be seen directly from the generating function since

$$\frac{e^{-t}}{1 - t} = \sum_{n \ge 0} \left( \sum_{k=0}^n \frac{(-1)^k}{k!} \right) t^n$$

which is equivalent to the formula $D_{n,0} = n! \sum_{k=0}^n \frac{(-1)^k}{k!} \sim \frac{n!}{e}$. (In fact you can read this asymptotic directly from the generating function.) The above then gives

$$D_{n,k} = {n \choose n-k} D_{n-k,0} = \frac{n!}{k!} \sum_{i=0}^{n-k} \frac{(-1)^i}{i!}.$$

Of course, there is a much more direct proof of this: observe that specifying a permutation of $n$ elements with $k$ fixed points is equivalent to specifying the $n-k$ elements which are not fixed points, then specifying a fixed-point-free permutation of these. This immediately gives $D_{n,k} = {n \choose n-k} D_{n-k,0}$, so it suffices to compute $D_{n,0}$, and this can be done by the standard inclusion-exclusion argument.

(In the interest of completeness, the standard inclusion-exclusion argument is as follows: first we start with all $n!$ permutations. Then we subtract the ones which fix $1$, and the ones which fix $2$, etc., so we subtract $n \cdot (n-1)!$. But this is overcounting: we need to add back the ones which fix both $1$ and $2$, or more generally both $i$ and $j$ for distinct $i, j$, so we add back ${n \choose 2} \cdot (n-2)!$. But this is overcounting: we need to subtract the ones which fix any triplet... and so forth. This gives each term of the formula $n! \sum_{k=0}^n \frac{(-1)^k}{k!}$ one-by-one.)

My point in presenting the generating function argument is not that it is any easier in this case but that it generalizes to far more complicated problems in a way which minimizes mental effort: for example you can use it to count permutations by how many $2$-cycles they have, or by $c_3(g) + 17 c_5(g)$, or whatever, and the generating function is also a convenient way to organize the computation of the expected value and variance of various permutation statistics; see, for example, this math.SE answer.

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Excellent explanation, thank-you! I didn't know about the Z_g identity, it may take a few reads, but I think I'm beginning to understand it now. –  Samonir Jan 13 '11 at 1:40
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The answer by Qiaochu Yuan is excellent, but perhaps admits some simplification. Note that the combinatorial species $\mathcal{Q}$ of permutations by cycles with the fixed points marked is given by $$\mathcal{Q} = \mathfrak{P}(\mathcal{U}\mathfrak{C_1}(\mathcal{Z}) + \mathfrak{C_2}(\mathcal{Z}) + \mathfrak{C_3}(\mathcal{Z}) +\cdots)$$ which immediately yields the generating function $$Q(z, u) = \exp\left(uz + \frac{z^2}{2} + \frac{z^3}{3} + \cdots\right) = \exp\left(uz -z + \log\frac{1}{1-z}\right) = \frac{1}{1-z} \exp\left(uz-z\right).$$ The coefficients from this generating function produce the rencontres numbers as in $$D_{n,k} = n! [z^n] [u^k] Q(z, u).$$ Actually doing the extraction we obtain $$D_{n,k} = n! [z^n] \frac{1}{1-z} \exp(-z) \frac{z^k}{k!} = \frac{n!}{k!} [z^{n-k}] \frac{1}{1-z} \exp(-z) = \frac{n!}{k!} \sum_{m=0}^{n-k} \frac{(-1)^m}{m!}.$$ Maybe this can contribute to an understanding of the essentials of this computation.

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