I came across one problem and I would like to find an answer. It is well-known how to calculate the number of fixed necklaces of length $n$ composed of $a$ types of beads $N(n,a)$. http://mathworld.wolfram.com/Necklace.html
So, let us consider very large necklaces with length $n$ and $n+1$. I wonder about the limit of $N(n+1,a)/N(n,a)$ for $n \to \infty$.
Actually I started from the finding the limit for the number of fixed necklaces for infinity. It seems to me that it should looks like a generating function for the total number of inversion in combinatorial theory and some factor which includes a power and factorial of $n$.
If you can make a computer experiment to see the numbers for very large $n$, it'll be a great opportunity to think about the numbers as well. Thank you for any help.