Weird Constants in Math Problems

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Weird constants in math problems

Michael Lugo recently considered a problem
involving the allocation of swimmers to swim lanes at random, ending

If we compute this for large !!n!! we get !!f(n) sim 0.4323n!!,
which agrees with the Monte Carlo simulations… The
constant !!0.4323!! is $$frac1{2(1-e^{-2})}.$$

I love when stuff like this happens. The computer is great at doing a
quick random simulation and getting you some weird number, and you
have no idea what it really means. But mathematical technique can
unmask the weird number and learn its true identity. (“It was Old Man
Haskins all along!”)

A couple of years back Math Stack Exchange had
Expected Number and Size of Contiguously Filled Bins,
and although it wasn’t exactly what was asked, I ended up looking into
this question: We take !!n!! balls and throw them at random into !!n!!
bins that are lined up in a row. A maximal contiguous sequence of
all-empty or all-nonempty bins is called a “cluster”. For example,
here we have 13 balls that I placed randomly into 13 bins:

13 boxes, some with blue balls.  The boxes
contain, respectively, 1, 0, 3, 0, 1, 2, 1, 1, 0, 1, 2, 1, 0 balls.

In this example, there are 8 clusters, of sizes 1, 1, 1, 1, 4, 1,
3, 1. Is this typical? What’s the expected cluster size?

It’s easy to use Monte Carlo methods and find that when !!n!! is
large, the average cluster size is approximately !!2.15013!!. Do you
recognize this number? I didn’t.

But it’s not hard to do the calculation analytically and discover that
that the reason it’s approximately !!2.15013!! is that the actual
answer is $$frac1{2(e^{-1} – e^{-2})}$$ which is approximately !!2.15013!!.

Math is awesome and wonderful.

(Incidentally, I tried the
Inverse Symbolic Calculator
just now, but it was no help. It’s also not in Plouffe’s
Miscellaneous Mathematical Constants)

(The expressions in the solutions of M. Lugo’s problem and this one
are very similar, and the two questions do seem related, but looking
at his analysis and at mine I see no reason why the answers should
have similar forms. I need to think about this more.)

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