Kipli's Cage

Swedish mathematician Lennart Carleson has been awarded the 2006 Abel Prize (BBC report). The award was based in large part on Carleson's work on Fourier series.

But I'm not sure what to make of this part of a Reuter's report:

"The prize statutes say it should aim to spur interest in mathematics among children and young people. So far, all winners of the prize have been in their 70s."

Is Reuter's trying to make a little dig at the Abel awards committee?

A recent post, Ask Mathforge: What are the Odds? at mathforge.net asks a question about the game of High-Low. In the game, a card is turned over (from the top of a regular shuffled deck) and the player is asked to guess if the next card will be higher or lower than the one shown. A simple strategy to play the game would be to guess 'High' if the card is an ace through 6 (consider ace to be '1'), 'Low' if the card is 8 through King, and flip a coin if the card is a 7. Intuitively, the player is playing the best he can without memory.

If we make the assumption that the player always gets the random coin flips correct, what is the probability that he will get every turn correct through the entire deck?

An estimate of the probability puts it at around 1.5 million to 1 (with a few other assumptions). In my approach to the problem, I wanted to count how many arrangements of the cards would be 'good' ones---the player will win following the given strategy (and luck to guess correctly on the middle 7 card). Since there are `{52!}/{(4!)^{13}}` possible arrangements of the cards, we could then determine the probability.

For ease of discussion, let's make the following definition. Given `n` cards of each rank from 1 to `m` (for a total of `mn` cards in the deck), a Hi-Lo arrangement of the cards is an arrangement `a_1 a_2 a_3 \dots a_{mn}` that satisfies

1. `a_i < a_{i+1}` if `1 \leq a_i < m/2`
2. `a_i > a_{i+1}` if `m/2 < a_i`, and
3. if `a_i = m/2` then there are no restrictions on `a_{i+1}`

Let `HL(m,n)` denote the number of Hi-Lo arrangements of cards of `m` ranks with `n` cards of each rank. We would like to find `HL(13,4)`.

However, computing `HL(13,4)` by hand (meaning: with a computer search), is not likely. The numbers grow very, very quickly. Here's a summary of some of the values for small `m` and `n`:

`HL(m,n)`	cards per rank
ranks	1	2	3	4
2	2	2	2	2
3	6	30	174	1092
4	14	230	4834	114442
5	78	14094	3785126	1225289412
6	230	187106	250560122
7	1902	26185806
8	6902	557115782
9	76110
10	329462

The even `m` values in the first column is sequence A048163 in the On-line Encyclopedia of Integer Sequences. The row corresponding to `m = 3` is sequence A110706.

Even for three ranks, the closed-form expression for `HL(3,n)` is complicated. I'm not even sure how to approach the problem of finding `HL(4,n)` for `n\geq 2`, or even what kinds of estimates would be reasonable for `HL(13,4)`.

So the question is still open: what is the probability of getting a HiLo arrangement?