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 $$\frac{1}{2^n} \sum_{i=0}^n (-1)^i \binom{n}{i} (2n-i)! 2^i$$

  • We have seen similar problems here before:… My impression is that there is no general formula for the solution, but I would love to be proven wrong! – user940 Apr 9 '12 at 2:30
  • quite different problem! – hkju Apr 9 '12 at 3:57
  • Not so different. – user940 Apr 9 '12 at 4:00
  • the problem you mentioned deals with the different number of cards, but here we consider the same number of cards. – hkju Apr 9 '12 at 6:41
  • The cases $k=2$ and $k=3$ are tabulated here: and . For $k=3$ there is no simple formula given. – user940 Apr 9 '12 at 14:17
up vote 31 down vote accepted

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 and 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!

  • I must be missing something. When $k=2$ and $n=1$, the correct answer is zero, but that's not what your formula gives. – user940 Apr 9 '12 at 22:36
  • Ahhh, sorry! I forgot a factor of $(-1)^{i-k}$ in the expression for $q_k$. I will fix it. – Jair Taylor Apr 10 '12 at 1:46
  • 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. – Jair Taylor Apr 10 '12 at 1:48
  • Here is some numerical evidence in the form of a Sage worksheet - it reproduces See – Jair Taylor Apr 10 '12 at 1:56
  • 1
    Well, I'm convinced! That is a wonderful formula. – user940 Apr 10 '12 at 1:59

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