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

[Neil Sloane]

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;
>
>
> --
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>