[Math] Hat Matching Problem Expectation

combinatoricspermutationsprobability

I have an interesting problem in the context of the hat matching problem:
There are $n$ people with hats at a party.
Each person randomly grabs a hat.
A match occurs if a person gets his own hat.

I'd like to know three things:

i) The probability that no one gets their hat

ii) How often we expect someone to get their hat

iii) If we let the n men choose their hats every minute, how much time do we expect until the first match (that is at minute 1, man 1 picks his hat, at minute 2, man 2 picks his hat, etc…)

I got number i) using inclusion exclusion, and ii) using linearity of expectation but I'm stuck on iii)

Here's what I've done so far.

Essentially we're trying to find the number of random permutations $\{x_1 , … , x_n\}$ of $\{1, … , n\}$ such that :

$x_i = i\;$ AND $\;x_j != j$ for all $j < i$

Since if we know this, then we know the probability of match occurring at any given time.

Intuitively, this is the same number of ways of getting $x_1 = 1, x_2 != 2, …, x_i != i$

But I can't see how to compute this since the number of ways of having $x_k != k$, for example, depends on wether or not $k$ was placed earlier in our permutation.

Thanks!

Best Answer

Here are some observations. Start with the species $\mathcal{Q}$ of permutations with fixed points marked, so that we have $$\mathcal{Q} = \mathfrak{P} \left(\mathcal{U}\mathcal{Z} + \mathfrak{C}_{=2}(\mathcal{Z}) + \mathfrak{C}_{=3}(\mathcal{Z}) + \mathfrak{C}_{=4}(\mathcal{Z}) + \cdots\right).$$ Translating to generating functions we obtain $$Q(z, u) = \exp\left( uz + \frac{z^2}{2} + \frac{z^3}{3} + \frac{z^4}{4} + \cdots \right) \\= \exp\left(uz -z + \log\frac{1}{1-z}\right) = \frac{e^{-z}}{1-z} e^{uz}.$$

The probability that no one gets their hat is computed from the count of permutations with no fixed points, $$D_n = n! [z^n] [u^0] Q(z, u) = n! [z^n] \frac{e^{-z}}{1-z} = n! \sum_{k=0}^n \frac{(-1)^k}{k!} \approx \frac{n!}{e}.$$ It follows that the probability is $1/e.$

For the expected value of people getting their hat we obtain $$[z^n] \left. \frac{d}{du} Q(z, u)\right|_{u=1} = [z^n] \left. \frac{z e^{-z}}{1-z} e^{uz}\right|_{u=1} = [z^n] \frac{z}{1-z} = 1$$ and thus there is an average of one fixed point per permutation.

As for the location of the smallest fixed point call it $q$ we can reason as follows. Take any permutation $\sigma$ of the $n-q$ elements that are larger than $q$ and factor it into cycles. Partition the $q-1$ elements smaller than $q$ into a derangement containing $p$ elements and distribute the remaining $q-1-p$ elements onto the cycles of the permutation $\sigma$ of $n-q$, which will assure that none of them becomes a fixed point and $q$ is the smallest fixed point because the elements less than $q$ are either in the derangement of $p$ elements or on the cycles of $\sigma,$ where none of them is a singleton.

This gives for $p$ fixed the value $$(n-q)! \times {q-1\choose p} \times D_p \times (n-q)(n-q+1)(n-q+2)\\ \cdots (n-q+(q-1-p-1)) \\= (n-q)! \times {q-1\choose p} \times D_p \times (n-q)(n-q+1)(n-q+2)\ldots (n-p-2) \\= (n-q)! \times {q-1\choose p} \times D_p \times {n-p-2\choose q-1-p} (q-1-p)!$$

When $q<n$ this can also be written as $$(n-q)! \times {q-1\choose p} \times D_p \times \frac{(n-p-2)!}{(n-q-1)!}$$ or $$(n-q) \times {q-1\choose p} \times D_p \times (n-p-2)!$$

The fourth term in the product above on the first line i.e. the product at the end represents the fact that we have $n-q$ choices when placing the first element from the $q-1-p$ elements on the cycles of $\sigma,$ and $n-q+1$ choices for the next one and so on. The first binomial coeffcient represents the choice of $p$ elements for the derangement.

The total contribution which is the desired sum of the smallest fixed point over all permutations is thus given by $$\sum_{q=1}^n q\times (n-q)! \times \sum_{p=0}^{q-1} {q-1\choose p} \times D_p \times {n-p-2\choose q-1-p} (q-1-p)!$$ or alternatively $$n\times D_{n-1} + \sum_{q=1}^{n-1} q\times (n-q)\times \sum_{p=0}^{q-1} {q-1\choose p} \times D_p \times (n-p-2)!$$ where we have used the fact that if $n$ is the first fixed point it must have been preceded by a derangement of the first $n-1$ elements.

This yields the sequence $$1, 1, 7, 31, 191, 1331, 10655, 95887, 958879, 10547659, \ldots$$

which points us to OEIS A155521 where additional material awaits and which would appear to confirm the above calculation.

The following admittedly simple Maple program was used to confirm the above formulae for small $n:$

P := proc(n)
local gf, p, pos;
option remember;
    gf := 0;
    for p in combinat:-permute(n) do
        for pos to n do if p[pos] = pos then break end if end do;
        if pos <= n then gf := gf + z^pos end if
    end do;
    gf
end proc

Addendum. (Inspired by the work of @YuvalFilmus.)

We learn from the OEIS entry that by a combinatorial bijection the sequence above is the number of permutations on $n+1$ elements having at least two fixed points. This has EGF $$Q_{\ge 2}(z) = \left. (Q(z, u) - [u^0] Q(z, u) - [u^1] Q(z, u))\right|_{u=1} = \frac{1}{1-z} - \frac{e^{-z}}{1-z} - \frac{ze^{-z}}{1-z}.$$

Extracting coefficients we get $$n! [z^n] Q_{\ge 2}(z) = n! \left(1 - \sum_{k=0}^n \frac{(-1)^k}{k!} - \sum_{k=0}^{n-1} \frac{(-1)^k}{k!}\right) \approx n! \left(1-\frac{2}{e}\right)$$

Dividing by the count of non-derangements we obtain $$\frac{(n+1)!(1-2/e)}{n!(1-1/e)} = (n+1) \times \frac{e-2}{e-1}.$$

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