[seqfan] Re: Modular Partitions

[Andrew Weimholt]

Hi Jens,

a while back I also looked at the idea of a modular version of the
partition numbers, but my definition is a bit different than yours.

See my comment in A002956 : Number of basic invariants for cyclic group of
order and degree n

a(n) is also the number of multisets of integers ranging from 1 to n, such
that the sum of the members of the multiset is congruent to 0 mod n, and no
submultiset exists whose sum of members is congruent to 0 mod n. These
multisets can be thought of as *partitions* of n in *modular* arithmetic,
thus this sequence can be thought of as a *modular* arithmetic version of
the *partition* numbers (Cf. A000041 <https://oeis.org/A000041>). - Andrew
*Weimholt*, Jan 31 2011

Also see my sequence A181887 : a(0) = 0 and A002956(n) - A000041(n) for n>0

which is (according to my definition of a modulo n partition) the number of
modulo n partitions of n which are not ordinary partitions of n.

My definition differs from yours in that I do not allow 0 to be an element
of the partition, and I do not
allow any submultiset which is also a modulo n partition of n.

Andrew


On Wed, Apr 30, 2014 at 1:03 PM, Neil Sloane <njasloane at gmail.com> 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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