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As mentioned in this question, the probability of winning, with an $n$-card per $k$ suit deck, a counting-up match card game (where you count through each of $n$ cards in order $k$ times and lose if you ever match the card drawn, i.e. you lose if card $i\equiv i \mod k$), is:

$$P(n,k)=\frac{1}{(nk)!}\intop_0^{\infty}x^{nk}R_k\left(-\frac{1}{x}\right)^{n}e^{-x}dx,$$

here $R$ represents a (square) rook polynomial

From my simulations it seems to be that $\lim\limits_{n\rightarrow\infty}P(n,k)$ is well-defined and nonzero for all fixed $k$; what is the limit? I've never been good with factorial limits, let alone with the Rook polynomials involved.

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MichaelChirico
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  • Why does the shuffle for $k=2$ have to be in two blocks of $4$? – joriki Jul 11 '15 at 22:29
  • Rats. I suppose it doesn't. Well, we have a lower bound... – MichaelChirico Jul 11 '15 at 22:35
  • I'm not convinced by the wlog argument either. I don't think it's clear that the situation is so symmetrical. On the $k=2$ part, it seems you would miss a solution like this: 2,1,1,2,3,4,4,3, just as one example of the previous commentator's point – Dr Xorile Jul 11 '15 at 22:40
  • I'm having trouble putting into words my reasoning for why I said WLOG, but I still think it is. – MichaelChirico Jul 11 '15 at 22:42
  • So you're saying that it's 3 x 3 = 9 that are safe out of 4! = 24 shuffles? 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321. Mmm. That does seem right. I wonder if you can really generalise then. Maybe you could figure it out recursively or by induction. – Dr Xorile Jul 11 '15 at 22:48
  • @joriki's observation has deflated my hopes in finding a general form. I still think there should be something closed form for $P(n,1)$--I feel it dancing just beyond my grasp. – MichaelChirico Jul 11 '15 at 22:50
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    So I stuck 0,1, 2, 9 into OEIS, and found this: http://oeis.org/A000166, the so-called derangement numbers, which rings a bell. – Dr Xorile Jul 11 '15 at 23:01
  • I added a new answer to the question that I marked this a duplicate of. – joriki Jul 12 '15 at 00:49

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