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In how many ways we can arrange N identical objects around a circle of size L. I think it would be somehow related to the number of ways of distributing n identical objects among r groups, where r=1 to N. But this is an arrangement problem, so the gap between clusters is important!

p.s. For those who are familiar with asymmetric simple exclusion process (ASEP) model in statistical mechanics, my question would be, the number of distinguish configurations for ASEP with N particles on a circle of size L.

Thanks,

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    This looks to be a necklace with two colors (included / not included), there is more at [MSE meta](http://meta.math.stackexchange.com/questions/1868/list-of-generalizations-of-common-questions#13335). Substitute $1+X$ (for off, on) into the cycle index $Z(C_L)$ of the cyclic group, expand, extract the coefficient on $X^N.$ Alternatively, extract using the binomial theorem. – 2017-02-22
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    @Marko Riedel, I so love Burnside's lemma / Pólya's enumeration theorem. – 2017-02-22
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    Great, Thank you so much Marko Riedel. – 2017-02-22
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    Here also is the answer http://math.stackexchange.com/questions/721783/number-of-unique-sequences-with-circular-shifts – 2017-02-22
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    Now, my next question is how large this number would be for large big and L, such that N/L is a constant. – 2017-02-22
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    The term $a_1^L/L$ dominates asymptotically in the substituted cycle index which says that most necklaces have no symmetry. These are polynomials of degree $L$ in the number $Q$ of colors being used. – 2017-02-22
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    Great help, Thanks a lot. – 2017-02-22

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