I am looking for an application of the Kendall-Mann sequence (KM) which uses the property $M(n+1)/M(n) = n - 1/2 + O(1/n)$ ($n \to \infty$) in science ( computer science ( sorting), physics, biology (genetics), etc) to use the property in some particular case with a specific error bound.

The property of Kendall-Mann numbers is proved, but not published in full details ( if you know the property, could you let me know the reference please).

To clarify the question: Kendall-Mann numbers M(n): the maximum number of permutations on n letters having the same number of inversions http://oeis.org/A000140 M(1)=1, M(2)=1, M(3)=2, M(4)=6, M(7)=22, M(8)=101… $M(n+1)/M(n)$ for n=1,… 29

$M(2)/M(1)=1$, $M(3)/M(2)=2$, $M(4)/M(3)=3$,...

1.00000000, 2.00000000, 3.00000000, 3.66666667, 4.59090909, 5.67326733, 6.69458988, 7.61939520, 8.57906801, 9.60953383, 10.6235009, 11.5884536, 12.5657349, 13.5817521, 14.5907723, 15.5704306, 16.5558579, 17.5656455, 18.5718445, 19.5585507, 20.5484134, 21.5549876, 22.5594838, 23.5501133, 24.5426559, 25.5473665, 26.5507683, 27.5438066, 28.5380914

It is proved that $M(n)=\frac {n!} {\sqrt{n(n-1)(2n+5)* \pi}} *(1+Q1/n+….)$, Q1- a constant. It presents the KM numbers. For more details please see The property of Kendall-Mann numbers

The question is what to do with the fact? Any applications in science just to show how it can be applied with a precise error?

anysequence which has the same property? In the first case it would be better to ask for "an application of the KM sequence which uses the property M(n+1)/M(n) = ..." and in the second case it would be better to ask for "an application of any sequence a(n) which has the property a(n+1)/a(n) = ..." – KConrad Jan 7 '11 at 16:26