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Stars and bars problem with the constraint that there has to be at least one bar between every star. Do we have formula for calculating the number of different cases?

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closed as unclear what you're asking by user21820, Jyrki Lahtonen, Brian Borchers, MR_BD, Daniel Sep 8 '18 at 22:25

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    $\begingroup$ There are $n!$ ways to permute $n$ (different) numbers, no matter what their sum is. $\endgroup$ – Henning Makholm Sep 17 '15 at 16:54
  • $\begingroup$ I have a feeling this is meant to be phrased as a "stars and bars with restrictions" problem. To that end, OP, would you know what to do if all of the terms could be any nonnegative natural number, and we forgot about the $k$? $\endgroup$ – pjs36 Sep 17 '15 at 16:58
  • $\begingroup$ @pjs36 yes, thank you, could you help edit the question? $\endgroup$ – talmikohujik Sep 17 '15 at 17:01
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You can look up "stars and bars" construction to find the number of positive integer solutions for a sum of $n$ positive integers to equal a target sum $S$. But if some numbers have to be at least $k$, then you can just subtract $k-1$ from each such number and subtract $k-1$ from the sum $S$ for each such number, and then you've reduced to the original stars and bars problem.

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It is pretty easy but I will just post it: Consider every star (except the last one) and the bar after it as one star (since the bar after the star is compulsory). Then we simplified our problem into a normal stars and bars one. Can be generalized if every star is followed by at least n bars.

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