It is known that quantum harmonic oscillator is connected to the symmetric group of infinite order which is isomorphic to the permutation group. According to Cayley's theorem any finite group is isomorphic to a subgroup of permutations of elements of this group.

Looking at Kendall-Mann numbers $M(n)$: the maximal number of inversions in a permutation on $n$ letters – we can see Kendall-Mann numbers’ property $M(n)/M(n-1)=n+\frac {1}{2}$ as $n \to \infty$. The same structure is for energy level of the quantum harmonic oscillator.

How can we explain this seemingly coincident based on mathematical approach? What is behind Hamilton operator $H$ and it's $\frac {1}{2}$ expressed as $H=\left(N+\frac {1}{2}\right)\frac {h}{2\pi} \omega$?