From: GeometricGroup on
> On 20 May, 14:54, GeometricGroup <ggx...(a)gmail.com>
> wrote:
> > The detailed Coxeter graph of my question is found
> in
> >
> >
> http://www.combinatorics.org/Volume_5/PDF/v5i1r18.pdf
> >
> > (In page 5, the first figure in table 1)
> >
> > I just started learning this stuff, so it confuses
> me a lot.
> >
> > Thanks for any help.
>
> You have removed the context!
>
> Properties 1 and 2 that you mentioned in your earlier
> post
> (corrected):
>
> > Let \tilde{S_n} be affine symmetric group. Elements
> x \in \tilde{S_n} are bijections from Z to itself
> satisfying:
> >
> > 1. x(i+n)=x(i)+n for all i \in Z.
> > 2. \sum_{i=1}^n x(i)=(n+1)C2, where nCr denotes "n
> choose r".
>
> are Lemma 4 and Proposition 9 of the paper you cited
> above.
>
> Derek Holt.
>

I was wondering where n+1C2 come from, but it turns out that it is simply the sum from 1 to n.

Thank you for your help.