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A330371
Irregular triangle read by rows T(n,m) in which row n lists all partitions of n ordered by the lower value of their k-th ranks, or by their k-th largest parts if all their k-th ranks are zeros, with k = n..1.
0
1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 4, 3, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 1, 4, 2, 4, 1, 1, 3, 3, 3, 2, 1, 2, 2, 2, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 6, 1, 5, 2, 5, 1, 1, 4, 3, 4, 2, 1, 3, 3, 1, 4, 1, 1, 1, 3, 2, 2, 3, 2, 1, 1
OFFSET
1,2
COMMENTS
In this triangle the partitions of n are ordered by their n-th rank. The partitions that have the same n-th rank appears ordered by their (n-1)-st rank. The partitions that have the same n-th rank and the same (n-1)-st rank appears ordered by their (n-2)-nd rank, and so on. The partitions that have all k-ranks equal zero appears ordered by their largest parts, then by their second largest parts, then by their third largest parts, and so on.
Note that a partition and its conjugate partition both are equidistants from the center of the list of partitions of n.
For further information see A330370.
First differs from A036037, A181317, A330370 and A334439 at a(48).
First differs from A080577 at a(56).
EXAMPLE
Triangle begins:
[1];
[2], [1,1];
[3], [2,1], [1,1,1];
[4], [3,1], [2,2], [2,1,1], [1,1,1,1];
[5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1];
[6], [5,1], [4,2], [4,1,1], [3,3], [3,2,1], [2,2,2], [3,1,1,1], [2,2,1,1], ...
.
For n = 9 the 9th row of the triangle contains the partitions ordered as shown below:
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Ranks
Conjugate
Label with label Partition k = 1 2 3 4 5 6 7 8 9
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1 30 [9] 8 -1 -1 -1 -1 -1 -1 -1 -1
2 29 [8, 1] 6 0 -1 -1 -1 -1 -1 -1 0
3 28 [7, 2] 5 0 -1 -1 -1 -1 -1 0 0
4 27 [7, 1, 1] 4 0 0 -1 -1 -1 -1 0 0
5 26 [6, 3] 4 1 -2 -1 -1 -1 0 0 0
6 25 [6, 2, 1] 3 0 0 -1 -1 -1 0 0 0
7 24 [6, 1, 1, 1] 2 0 0 0 -1 -1 0 0 0
8 23 [5, 4] 3 2 -2 -2 -1 0 0 0 0
9 22 [5, 3, 1] 2 1 -1 -1 -1 0 0 0 0
10 21 [5, 2, 2] 2 -1 1 -1 -1 0 0 0 0
11 20 [5, 2, 1, 1] 1 0 0 0 -1 0 0 0 0
12 19 [4, 4, 1] 1 2 -1 -2 0 0 0 0 0
13 18 [4, 3, 2] 1 0 0 -1 0 0 0 0 0
14 17 [4, 3, 1, 1] 0 1 -1 0 0 0 0 0 0
15 (self-conjugate) [5, 1, 1, 1, 1] All zeros -> 0 0 0 0 0 0 0 0 0
16 (self-conjugate) [3, 3, 3] All zeros -> 0 0 0 0 0 0 0 0 0
17 14 [4, 2, 2, 1] 0 -1 1 0 0 0 0 0 0
18 13 [3, 3, 2, 1] -1 0 0 1 0 0 0 0 0
19 12 [3, 2, 2, 2] -1 -2 1 2 0 0 0 0 0
20 11 [4, 2, 1, 1, 1] -1 0 0 0 1 0 0 0 0
21 10 [3, 3, 1, 1, 1] -2 1 -1 1 1 0 0 0 0
22 9 [3, 2, 2, 1, 1] -2 -1 1 1 1 0 0 0 0
23 8 [2, 2, 2, 2, 1] -3 -2 2 2 1 0 0 0 0
24 7 [4, 1, 1, 1, 1, 1] -2 0 0 0 1 1 0 0 0
25 6 [3, 2, 1, 1, 1, 1] -3 0 0 1 1 1 0 0 0
26 5 [2, 2, 2, 1, 1, 1] -4 -1 2 1 1 1 0 0 0
27 4 [3, 1, 1, 1, 1, 1, 1] -4 0 0 1 1 1 1 0 0
28 3 [2, 2, 1, 1, 1, 1, 1] -5 0 1 1 1 1 1 0 0
29 2 [2, 1, 1, 1, 1, 1, 1, 1] -6 0 1 1 1 1 1 1 0
30 1 [1, 1, 1, 1, 1, 1, 1, 1, 1] -8 1 1 1 1 1 1 1 1
CROSSREFS
Another version of A330370.
Row n contains A000041(n) partitions.
Row n has length A006128(n).
The sum of n-th row is A066186(n).
For the "k-th rank" see also: A181187, A208478, A208479, A208482, A208483.
Sequence in context: A036037 A181317 A330370 * A080577 A374515 A302246
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Dec 15 2019
STATUS
approved