Find the number of arrangements of $k \mbox{ }1'$s, $k \mbox{ }2'$s, $\cdots, k \mbox{ }n'$s - total $kn$ cards - so that no same numbers appear consecutively. For $k=2$ we can compute it by using the PIE, and it is $$\frac{1}{2^n} \sum_{i=0}^n (-1)^i \binom{n}{i} (2n-i)! 2^i$$
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$\begingroup$ We have seen similar problems here before: math.stackexchange.com/questions/76213/… My impression is that there is no general formula for the solution, but I would love to be proven wrong! $\endgroup$– user940Commented Apr 9, 2012 at 2:30
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$\begingroup$ quite different problem! $\endgroup$– hkjuCommented Apr 9, 2012 at 3:57
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$\begingroup$ Not so different. $\endgroup$– user940Commented Apr 9, 2012 at 4:00
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$\begingroup$ the problem you mentioned deals with the different number of cards, but here we consider the same number of cards. $\endgroup$– hkjuCommented Apr 9, 2012 at 6:41
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$\begingroup$ The cases $k=2$ and $k=3$ are tabulated here: oeis.org/A114938 and oeis.org/A193638 . For $k=3$ there is no simple formula given. $\endgroup$– user940Commented Apr 9, 2012 at 14:17
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1 Answer
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I believe the answer is given by $$\int_0^\infty e^{-x} q_k(x)^n \, dx$$ where $q_k(x) = \sum_{i=1}^k \frac{(-1)^{i-k}}{i!} {k-1 \choose i-1}x^i$ for $k\geq 1$, and $q_0(x) = 1$. In general if we allow $k_i$ of the $i$th number the answer should be $$\int_0^\infty e^{-x} \prod_i q_{k_i}(x) \, dx$$
You can check that this agrees with the sequences oeis.org/A114938 and oeis.org/A193638 above. I do not (quite) have a proof of this, although I'm very close. The method is my own, and has not been published anywhere as far as I know. I'd be happy to give you more information in private, but I'm not sure I want to expose it publicly until it's proven. Please let me know if you think this is noteworthy and any potential applications.
Edit: Following some information given to me by Byron, I found that this formula is already known and that in fact $q_n(x) = (-1)^{n}L_n^{(-1)}(x)$ where $L_n^{(\alpha)} (x) $ denotes the generalized Laguerre polynomial. See Section 6 here for a labelled version. I should have mentioned this sooner; thanks Byron!
answered Apr 9, 2012 at 21:42
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$\begingroup$ I must be missing something. When $k=2$ and $n=1$, the correct answer is zero, but that's not what your formula gives. $\endgroup$– user940Commented Apr 9, 2012 at 22:36
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$\begingroup$ Ahhh, sorry! I forgot a factor of $(-1)^{i-k}$ in the expression for $q_k$. I will fix it. $\endgroup$ Commented Apr 10, 2012 at 1:46
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$\begingroup$ You can check that $q_2(x) = -x + \frac{x^2}{2}$ and then $\int_0^\infty e^{-x} ( -x + \frac{x^2}{2}) \, dx = 0$ as desired. $\endgroup$ Commented Apr 10, 2012 at 1:48
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1$\begingroup$ Well, I'm convinced! That is a wonderful formula. $\endgroup$– user940Commented Apr 10, 2012 at 1:59
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1$\begingroup$ Yes - I don't know if this helps, but $q_k(x) = \int_0^x q_{k-1}(t)\, dt - q_{k-1}(x)$. Also, $q_k(x)$ is the inverse Laplace of $(1-x)^{k-1}/x^{k+1}$. Can I ask why this question came up? $\endgroup$ Commented Apr 11, 2012 at 1:14