[seqfan] Re: The number of orbits of triples of (1, 2, ..., n) under the action of the dihedral group of order 2n

[Neil Sloane]

It should not be difficult to prove that that generating
function is correct using Polya's counting theory.
You just need the cycle index for that action of D_2n.
Magma and Mathematica both have commands
for finding the cycle index.

Neil


On Tue, Jan 21, 2014 at 1:17 PM, W. Edwin Clark <wclark at mail.usf.edu> wrote:

> I just submitted this sequence (A236283) which I was surprised to not find
> in the OEIS. It is an analogue of a question that arose for the action of
> another group.  It doesn't seem to fit any of the necklace sequences.
> Perhaps someone can think of a nice geometric interpretation.
>
> The number of orbits of triples of (1,2,...,n) under the action of the
> dihedral group of order 2n
>
> 1, 4, 5, 10, 13, 20, 25, 34, 41, 52, 61, 74, 85, 100, 113, 130, 145, 164,
> 181, 202, 221, 244, 265, 290, 313, 340, 365, 394, 421, 452, 481, 514, 545,
> 580, 613, 650, 685, 724, 761, 802, 841, 884, 925, 970, 1013, 1060, 1105,
> 1154, 1201, 1252
>
> EXAMPLE
> For n = 3 there are 5 orbits of triples:
> [ [ 1, 1, 1 ], [ 2, 2, 2 ], [ 3, 3, 3 ] ]
> [ [ 1, 1, 2 ], [ 2, 2, 3 ], [ 1, 1, 3 ], [ 3, 3, 1 ], [ 3, 3, 2 ], [ 2, 2,
> 1 ] ]
> [ [ 1, 2, 1 ], [ 2, 3, 2 ], [ 1, 3, 1 ], [ 3, 1, 3 ], [ 3, 2, 3 ], [ 2, 1,
> 2 ] ]
> [ [ 1, 2, 2 ], [ 2, 3, 3 ], [ 1, 3, 3 ], [ 3, 1, 1 ], [ 3, 2, 2 ], [ 2, 1,
> 1 ] ]
> [ [ 1, 2, 3 ], [ 2, 3, 1 ], [ 1, 3, 2 ], [ 3, 1, 2 ], [ 3, 2, 1 ], [ 2, 1,
> 3 ] ]
>
>
> (GAP)
> a:=function(n)
> local g, orbs;
> g:=DihedralGroup(IsPermGroup, 2*n);
> orbs := OrbitsDomain(g, Tuples( [ 1 .. n ], 3), OnTuples );
> return Size(orbs);
> end;;
>
> Maple's guessgf gives this generation function:
> -(2*x^3-3*x^2+2*x+1)/(x^4-2*x^3+2*x-1)
>
> --Edwin
>
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>
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>



-- 
Dear Friends, I have now retired from AT&T. New coordinates:

Neil J. A. Sloane, President, OEIS Foundation
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
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