[seqfan] Re: Modular Partitions

[Roland Bacher]

The link with Molien series is easy to explain:

Molien series count dimensions of polynomials invariant 
under the action of a linear group. 

Cyclic groups act diagonally by multiplication with a root of 1 
(of order divisible by N for Z/NZ).

Consider now the action of Z/NZ on R[x_0,..x_{N-1}]
acting by multiplication with $e^{2ij\pi/N} on x_j.

The dimension of invariant polynomials of homogeneous degree
k is then exactly the modular partition number n,k 
mentionned by Jens Voss

Molien series are rational and can be computed using a determinantal
formula. This should allow to prove the observed symetry.

Roland



On Wed, Apr 30, 2014 at 04:03:46PM -0400, Neil Sloane wrote:
> A brief look at the triangle suggests that row n gives
> the coeffts in the Molien series for the cyclic group of order n.
> See A007997, A008610, A008646, A032191.
> If true this should follow immediately from the definition.
> A good reference is B. Sturmfels, Algorithms in Invariant Theory, Springer,
> '93, p. 65.
> 
> I don't have time to look into this further right now.
> 
> Neil
> 
> 
> 
> On Wed, Apr 30, 2014 at 3:19 PM, Jens Voß <jens at voss-ahrensburg.de> wrote:
> 
> >
> > Hi there, sequence fans,
> >
> > I was playing around with what I call "modular partition numbers":
> > Essentially different ways to write the neutral element of the group Z/nZ
> > as a sum of length k (for given n, k > 0).
> >
> > For example, for n = 5 and k = 4, we have thepartitions
> >
> > 0+0+0+0 = 0
> > 0+0+1+4 = 5 = 0
> > 0+0+2+3 = 5 = 0
> > 0+1+1+3 = 5 = 0
> > 0+1+2+2 = 5 = 0
> > 0+2+4+4 = 10 = 0
> > 0+3+3+4 = 10 = 0
> > 1+2+3+4 = 10 = 0
> > 1+3+3+3 = 10 = 0
> > 3+4+4+4 = 15 = 0
> >
> > so the number of 5-modular partitions of length 4 is 10.
> >
> > I computed the the values for n + k < 20 (as a square array read by
> > antidiagonals), and was somewhat surprised that this sequence isn't yet in
> > the database (even though several of the rows resp. columns are). However,
> > I was even more surprised to find that the array is symmetric in n and k:
> >
> > 1    1    1    1    1    1    1    1    1    1    1    1    1 1    1    1
> >    1    1    1
> > 1    2    2    3    3    4    4    5    5    6    6    7    7 8    8    9
> >    9   10
> > 1    2    4    5    7   10   12   15   19   22   26   31   35 40   46   51
> >   57
> > 1    3    5   10   14   22   30   43   55   73   91  116  140 172  204  245
> > 1    3    7   14   26   42   66   99  143  201  273  364  476 612  776
> > 1    4   10   22   42   80  132  217  335  504  728 1038 1428 1944
> > 1    4   12   30   66  132  246  429  715 1144 1768 2652 3876
> > 1    5   15   43   99  217  429  810 1430 2438 3978 6310
> > 1    5   19   55  143  335  715 1430 2704 4862 8398
> > 1    6   22   73  201  504 1144 2438 4862 9252
> > 1    6   26   91  273  728 1768 3978 8398
> > 1    7   31  116  364 1038 2652 6310
> > 1    7   35  140  476 1428 3876
> > 1    8   40  172  612 1944
> > 1    8   46  204  776
> > 1    9   51  245
> > 1    9   57
> > 1   10
> > 1
> >
> > I haven't been able to come up with a formula for the numbers (neither
> > recursive nor direct), and I don't see an immediate reason for the symmetry
> > either (some sort of dualism). Can somebody find a formula or explain why
> > the array is symmetric?
> >
> > Best regards,
> > Jens
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
> 
> 
> 
> -- 
> 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.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
> 
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