[seqfan] Re: Modular Partitions
Chris
cgribble263 at btinternet.com
Thu May 1 00:05:55 CEST 2014
In the example, why are the following partitions not present?
1+1+1+2 = 5 == 0 mod 5
1+1+4+4 = 10 == 0 mod 5
2+2+2+4 = 10 == 0 mod 5
2+2+3+3 = 10 == 0 mod 5
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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