The problem of Necklaces is well-known, i.e "The number of fixed necklaces of length $n$ composed of $a$ types of beads $N(n,a)$" can be calculated: http://mathworld.wolfram.com/Necklace.html

Let us consider the limit $\lim_{n\to \infty}\prod_{p=1}^n N(p,a)$.

It is possible to show that the limit presents the result which looks like the generating function for inversion ( we may exclude one unimportant factor):
$\frac {a^n} {n!}$ $\prod_{p=1}^n \frac {1-a^p} {1-a}$

For $n \to \infty$ we have $\prod_{p=1}^n N(p,a) \approx \frac {a^n} {n!}$ $\prod_{p=1}^n \frac {1-a^p} {1-a}$. Then, for eg please see theorem #1 http://www.cs.uwaterloo.ca/journals/JIS/VOL4/MARGOLIUS/inversions.pdf The generating function under theorem 1 looks like $\prod_{p=1}^n \frac {1-a^p} {1-a}$

So, a question appears, how to explain the influence of the symmetric group's properties for the particular case? In other words why and how the connection appears?

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