[seqfan] Re: easy? graph question

[Benoît Jubin]

Dear Neil,

I had a little trouble understanding your definition, in particular
the phrase "maximal number of nodes that are separated by at least
three edges".

As a mathematician not familiar with codes, I would propose : "a(n) is
the maximal cardinality of a set of such n-tuples whose members are at
distance at least three from each other" (or something similar, in
better English!).

Best regards,
Benoît








On Wed, Mar 15, 2017 at 7:25 PM, Neil Sloane <njasloane at gmail.com> wrote:
> Dear Seq Fans,  Someone just asked me to explain A085680, which is:
>
>
> Size of largest code of length n and constant weight 2 that can
> correct a single adjacent transposition.  The data line gives only
> these values, with offset 2:
>
> 1, 1, 2, 3, 4, 6, 7, 9, 11, 13, 15, 17, 20, 23, 26, 29, 32, 36, 40,
> 44, 48, 52, 57, 62
>
> I just added an explanation, as follows:
>
> Look at the set of n-choose-2 binary vectors of length n with exactly
> two 1's and n-2 0's in each vector.
>
> If we can get a vector v from a vector u by swapping 2 adjacent
> coordinates, say that u and v are adjacent.
>
> a(n) is the maximal number of nodes in the resulting graph that are
> separated by at least three edges.
>
> For n=4 the graph is
>
> .................1001
>
> ................../....\
>
> 1100--1010......0101--0011
>
> ..................\..../
>
> ..................0110
>
> so a(4) = 2 (use 1100 and 0011 as the code, or 1100 and 0101).
>
> More terms, anyone?
>
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