[seqfan] Re: Modular Partitions

[franktaw at netscape.net]

The sequence for necklaces, offset 0, is in A047996. I have created 
A241926, which is the table of the number of necklaces, offset 1. I 
think A047996 ought to be the main sequence for this.

I'm holding off on tying A047996 to Jens's problem. I still don't see a 
proof of equivalence, although the numerical evidence is convincing. If 
no proof is forthcoming soon, I will put it in as a conjecture.

Franklin T. Adams-Watters

-----Original Message-----
From: franktaw <franktaw at netscape.net>
To: seqfan <seqfan at list.seqfan.eu>
Sent: Thu, May 1, 2014 9:03 pm
Subject: [seqfan] Re: Modular Partitions


Number of necklaces with n+k beads, n white and k black? I'm pretty
sure that's right, though I don't see how to prove it.

Franklin T. Adams-Watters

-----Original Message-----
From: Jens Voß <jens at voss-ahrensburg.de>

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

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