'Mixing number' of permutations

[Henry Gould]

Ah yes, here it is:

A046879 <http://www.research.att.com/%7Enjas/sequences/A046879> 	
	Denominator of 1/n Sum[ 1/BinomialCoefficient[ n-1,k ], {k,0,n-1} ]. 	
	+20
4


	1, 1, *1, 6, 3, 15, 30, 420*, 105, 315, 315, 6930, 3465, 90090, 180180, 
72072, 9009, 153153, 153153, 5819814, 14549535, 14549535, 29099070, 
1338557220, 334639305, 1673196525, 1673196525, 10039179150, 10039179150, 
582272390700, 1164544781400 (list 
<http://www.research.att.com/%7Enjas/sequences/table?a=46879&fmt=4>; 
graph <http://www.research.att.com/%7Enjas/sequences/table?a=46879&fmt=5>)


	OFFSET 	

0,4

	

	LINKS 	

Eric Weisstein's World of Mathematics, Leibniz Harmonic Triangle 
<http://mathworld.wolfram.com/LeibnizHarmonicTriangle.html>


	CROSSREFS 	

See A046825 <http://www.research.att.com/%7Enjas/sequences/A046825>, the 
main entry for this sequence. Cf. A046878 
<http://www.research.att.com/%7Enjas/sequences/A046878>.

Sequence in context: A049784 
<http://www.research.att.com/%7Enjas/sequences/A049784> A097917 
<http://www.research.att.com/%7Enjas/sequences/A097917> A116570 
<http://www.research.att.com/%7Enjas/sequences/A116570> this_sequence 
A067990 <http://www.research.att.com/%7Enjas/sequences/A067990> A050008 
<http://www.research.att.com/%7Enjas/sequences/A050008> A019069 
<http://www.research.att.com/%7Enjas/sequences/A019069>

Adjacent sequences: A046876 
<http://www.research.att.com/%7Enjas/sequences/A046876> A046877 
<http://www.research.att.com/%7Enjas/sequences/A046877> A046878 
<http://www.research.att.com/%7Enjas/sequences/A046878> this_sequence 
A046880 <http://www.research.att.com/%7Enjas/sequences/A046880> A046881 
<http://www.research.att.com/%7Enjas/sequences/A046881> A046882 
<http://www.research.att.com/%7Enjas/sequences/A046882>


	KEYWORD 	

nonn,frac,easy,nice


	AUTHOR 	

njas


= = = = = = =
and: 

	
	Numerator of Sum_{k=0..n} 1/C(n,k). 	
	+0
14


	1, 2, 5, 8, 8, 13, 151, 256, 83, 146, 1433, 2588, 15341, 28211, 52235, 
19456, 19345, 36362, 651745, 6168632, 1463914, 2786599, 122289917, 
233836352, 140001721, 268709146, 774885169, 1491969394, 41711914513, 
80530073893 (list 
<http://www.research.att.com/%7Enjas/sequences/table?a=46825&fmt=4>; 
graph <http://www.research.att.com/%7Enjas/sequences/table?a=46825&fmt=5>)


	OFFSET 	

0,2

	

	REFERENCES 	

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294, Problem 7.15.

R. L. Graham, D. E. Knuth, O. Patashnik; Concrete Mathematics, 
Addison-Wesley, Reading (1994) 2nd Ed. Exercise 6.100.

G. Letac; Problemes de probabilites, Presses Universitaires de France 
(1970), p. 14

F. Nedemeyer and Y. Smorodinsky, Resistances in the multidimensional 
cube, Quantum 7:1 (Sep./Oct. 1996) 12-15 and 63.

D. Singmaster, SIAM Review, 22 (1980) 504, Problem 79-16, Resistances in 
an n-Dimensional Cube.

B. Sury, Sum of the reciprocals of the binomial coefficients, Europ. J. 
Combinatorics, 14 (1993), 351-353.


Do there are references to the fractions i  OEIS already. Summing the 
reciprocals of the binomial coefficients was one of the early problems I 
got into when I began to study
binomial sums in 1945. I should have listed this in my 1959 report and 
in my 1972 book of 500 binomial sums.

Regards,

Henry Gould

Brendan McKay wrote:
> The average is  sum(1/binomial(n,k),k=1..n-1).  I don't know
> if this has a closed form or a name.
>
> Proof:  There are clearly k! (n-k)! permutations that have a 
> break after the k-th value, so the probability of a break in
> that place is 1/binomial(n,k).
>
> Asymptotically, the values k=1 and k=n-1 dominate, so the
> expectation is 2/n + O(1/n^2).
>
> Brendan.
>
>
> * Hugo Pfoertner <all at abouthugo.de> [060817 22:30]:
>   
>> SeqFans,
>>
>> today Hauke Reddmann started a new thread in the NG sci.math
>>
>> http://groups.google.com/group/sci.math/msg/b1c1164b936c6dca
>>
>> <<
>> For easyness, I use the data of my lest chess tournament :-)
>> The finish in terms of the starting numbers was
>> 7 1 6 4 3 8 5 2|9|10|14 15 20 13 18 16 23 22 12 26 11 17 19 27 25 21
>> 24|28
>>
>> A | marks boundaries between consecutive number subsets that permute
>> to themselves. Note that I (the 16) also permute to myself, but there
>> are number crossing from both sides and so this is no boundary.
>>
>> Obvious question: How many boundaries occur in a random permutation?
>> Clearly a tournament is about the opposite of random, as the swap
>> numbers will be low.
>>
>> Example n=3
>>
>> 1|2|3
>> 1|3 2
>> 2 1|3
>> 2 3 1
>> 3 1 2
>> 3 2 1
>> -- 
>> Hauke Reddmann <:-EX8    fc3a501 at uni-hamburg.de
>> His-Ala-Sec-Lys-Glu Arg-Glu-Asp-Asp-Met-Ala-Asn-Asn
>>     
>> Is there anything in the OEIS that can answer his question? Any other
>> ideas?
>>
>> Can you please CC answers also to Hauke; AFAIK he is not member of the
>> Seqfan community.
>>
>> Hugo Pfoertner
>>     
>
>