[seqfan] Re: Modular Partitions

[jean-paul allouche]

Hi, I suspect the following paper might help:
http://dx.doi.org/10.1016/0012-365X(86)90089-0

best wishes
jean-paul


Le 02/05/14 04:03, franktaw at netscape.net a écrit :
> 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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