# nim with a difference

Antti Karttunen karttu at megabaud.fi
Wed Jan 23 22:50:02 CET 2002

```
Marc LeBrun wrote:

>  >=Antti Karttunen
>  > [...]
>  >  2,11,35,85,175,322,546,870,1320,1925,2717,3731,5005,6580,
>  > s(n+2,n), that is the number of  n+2 letter permutations with n cycles.
>  > So only cycles that can occur are 1-, 2- and 3-cycles.
>
> OK, I *think* I understand your argument:  Since after the 2nd move you can
> only have these kinds of cycles then the case analysis can be carried out
> explicitly (as you proceed to do).

Yes.

>
>
> So then what is the next (3rd move) anti-diagonal 4, 40, 195, 665...?  Do
> these diagonal sequences form some family that goes
>
>    constant 1, choose(n,2)=choose(n,n-2), stirling(n+2,n), ...?

Who would like to make an explicit list of cases like
| | | a - b | - c |  - d | and so on, and see how many
equivalence classes there are? And then the 4th move,
and see whether any pattern emerges.
Symbolic programming language like Maple
or Mathematica might be useful, if one could
specify for it the relative order of "uninstantiated"
variables a, b, c, d, etc, and conditions like a >= b+c+d,
etc.

> "Enjoy"!

Yes, I did like this one. Thanks!

Yours,

Antti

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