[seqfan] Re: Modular Partitions

[Chris]

It appears that T(n, k) = (n + k - 1)! / (k! * n!) for all n == +-1 mod k, k > 1.
In addition, it appears that
T(n, 2) = ((n + 1) + 1) / 2!, n == 0 mod 2 (A000012)
T(n, 3) = ((n + 2)(n + 1) + 4) / 3!, n == 0 mod 3 (A007997)
T(n, 4) = ((n + 3)(n + 2)(n + 1) + 3 * (n + 6)) / 4!, n == 0 mod 4; ((n + 3)(n + 2)(n + 1) + 3 * (n + 2)) / 4!, n == 2 mod 4 (A008610)
T(n, 5) = ((n + 4)(n + 3)(n + 2)(n + 1) + 96) / 5!, n == 0 mod 5; (n + 4)(n + 3)(n + 2)(n + 1) / 5!, n == 2:3 mod 5 (A008646)

Conjecture: For p prime,
T(n, p) = (n + p - 1)! / (p! * n!) for n !== 0 mod p.
T(n, p) = (n + p - 1)! / (p! * n!)  + (p - 1) / p for n == 0 mod p.

Best regards,
Chris Gribble

-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Jens Voß
Sent: 30 April 2014 20:19
To: Sequence Fanatics Discussion list
Subject: [seqfan] Modular Partitions


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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