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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?

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Folks might want to look at Mikhail's earlier question about these numbers, mathoverflow.net/questions/46368/… –  Gerry Myerson Jan 7 '11 at 2:47
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You are asking for an application of an asymptotic property of this sequence. Sorry to be pedantic, but are you looking for an application of this specific sequence which uses that property or are you looking for an application of any sequence 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
    
This is an odd sort of question: "Here is a theorem so new that it's not even published yet; can you find something to apply it to?" I can see why people are voting to close. –  Gerry Myerson Jan 8 '11 at 0:23
    
Well, could you advise me what the question to ask about the property please? I would not like to ask odd questions. If the property is uknown, then I would publish it in a paper adding a particular case with a specific error bound. I know the property, but I know nothing where to apply it –  Mikhail Gaichenkov Jan 8 '11 at 9:46
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