'Mixing number' of permutations

[Henry Gould]

If I am not mistaken the number

sum(1/binomial(n,k),k=1..n-1)  is the diagonal resistance in ohms of an n-dimensional cube, each edge of which is a 1 ohm resistor. K had to figure this out using Kirchhoff's laws back in 1948. 

Can someone concur with this????

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
>>     
>
>