[seqfan] Re: Partition into Stroke of the star graph

[Neil Sloane]

I added a comment based on Max's reply.

Best regards
Neil

Neil J. A. Sloane, Chairman, OEIS Foundation.
Also Visiting Scientist, Math. Dept., Rutgers University,
Email: njasloane at gmail.com



On Thu, May 4, 2023 at 11:56 AM Max Alekseyev <maxale at gmail.com> wrote:

> As I understand from Christian messages, there is no concern about
> correctness of A089243, except for the case n=1.
> I agree that value for n=1 can be disputable, but this really boils down to
> what we understand under the star graph in this case. It is clearly the
> graph composed just of a single edge connecting two vertices. If one of
> them is labeled as the star center, then we have the current
> value A089243(1)=2. However, if the center is not distinguished, then the
> value would be 1.
> Perhaps, it's worth adding a comment to the sequence explaining this issue.
>
> Regards,
> Max
>
>
> On Thu, May 4, 2023 at 11:13 AM Neil Sloane <njasloane at gmail.com> wrote:
>
> > Concerning A089243, I will ask Max Alekseyev to reply.
> >
> > Best regards
> > Neil
> >
> > Neil J. A. Sloane, Chairman, OEIS Foundation.
> > Also Visiting Scientist, Math. Dept., Rutgers University,
> > Email: njasloane at gmail.com
> >
> >
> >
> > On Wed, May 3, 2023 at 11:40 PM Christian Sievers <seqfan at duvers.de>
> > wrote:
> >
> > > Some remarks and results:
> > >
> > > 1) The value of A089243(1)=2 can be disputed.
> > >
> > > We can only distinguish the two stroke sets for a star graph with one
> > > edge if the center node is distinguished. In all other cases (even n=0)
> > > it is obvious which point is the center.
> > >
> > > 2) In the mail, what does this mean?
> > >
> > > >     1, 2, 3, 4, 9, 22, 61, 200,    Off SET 0    0 means Phi
> > >                                                   ^^^^^^^^^^^
> > > It's not from the OEIS entry.
> > >
> > > 3) Also from the mail:
> > >
> > > >     The definition is for the case of directed , labelled , mirror
> > > > symmetric exist    I computed  all cases  that exist eight
> > >
> > > We can certainly consider variations of the series, like undirected
> > > strokes (then we act on sets of sets of size 2, instead of sets of
> > > pairs, and count each 2-stroke possibility only once)
> > > and/or only consider stroke sets equivalent if one can be obtained from
> > > the other by a rotation and not allow reflections (so replace the
> > > dihedral group by a cyclic group),
> > > but I don't see what labeled vs unlabeled should mean.
> > > Without labels I can't differentiate the nodes, and then the symmetries
> > > become meaningless.
> > >
> > > I computed the cases that make sense to me, here are the results
> > > (code at the end of the mail):
> > >
> > >   # as before
> > >   gap> List([1..10],i->f(i,true,true));
> > >   [ 2, 3, 4, 9, 22, 61, 200, 689, 3054, 12110 ]
> > >
> > >   # only rotations are equivalent
> > >   gap> List([1..10],i->f(i,true,false));
> > >   [ 2, 3, 6, 12, 34, 98, 374, 1290, 5962, 23768 ]
> > >
> > >   # undirected
> > >   gap> List([1..10],i->f(i,false,true));
> > >   [ 1, 2, 2, 5, 6, 17, 27, 83, 185, 608 ]
> > >
> > >   # undirected and only rotations
> > >   gap> List([1..10],i->f(i,false,false));
> > >   [ 1, 2, 2, 5, 6, 18, 34, 108, 294, 984 ]
> > >
> > > One might argue that in the undirected case the maximal stroke
> condition
> > > allows at most one 1-stroke. Then the results (code not given here) are
> > >   [ 1, 1, 1, 2, 3, 5, 11, 17, 65, 79 ]
> > > with reflection and rotations and
> > >   [ 1, 1, 1, 2, 3, 5, 15, 18, 105, 105 ]
> > > with only rotations considered equivalent.
> > >
> > > Of these, only the original series is in the OEIS.
> > >
> > >
> > > Best
> > > Christian
> > > --
> > >
> > > f := function(n,dir,refl)
> > >   local s, g, mult, act, h, res;
> > >
> > >   if dir then
> > >     mult := 2;
> > >     act := OnSetsTuples;
> > >   else
> > >     mult := 1;
> > >     act := OnSetsSets;
> > >   fi;
> > >
> > >   s := SymmetricGroup( n );
> > >
> > >   if refl then
> > >     g := DihedralGroup( IsPermGroup, 2*n);
> > >   else
> > >     g := CyclicGroup( IsPermGroup, n);
> > >   fi;
> > >
> > >   h := function( l )
> > >     local tss,x;
> > >     tss := Set([1..l], i->[2*i-1, 2*i]); # a set of l two-strokes
> > >     x := Orbit( s, tss, act ); # all such sets
> > >     return Size( OrbitsDomain( g, x, act ) );
> > >   end;
> > >
> > >   res := mult * Sum( [0..QuoInt(n-1,2)], h );
> > >   if IsEvenInt(n) then
> > >     res := res + h( n/2 );
> > >   fi;
> > >   return res;
> > > end;
> > >
> > >
> > > --
> > > Seqfan Mailing list - http://list.seqfan.eu/
> > >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
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