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A214314
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Number triangle with entry T(n,m) giving the position of the first partition of n with m parts in the Abramowitz-Stegun (A-St) partition ordering.
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6
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1, 1, 2, 1, 2, 3, 1, 2, 4, 5, 1, 2, 4, 6, 7, 1, 2, 5, 8, 10, 11, 1, 2, 5, 9, 12, 14, 15, 1, 2, 6, 11, 16, 19, 21, 22, 1, 2, 6, 13, 19, 24, 27, 29, 30, 1, 2, 7, 15, 24, 31, 36, 39, 41, 42, 1, 2, 7, 17, 28, 38, 45, 50, 53, 55, 56, 1, 2, 8, 20, 35, 48, 59, 66, 71, 74, 76, 77
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OFFSET
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1,3
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COMMENTS
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For the Abramowitz-Stegun ordering of partitions see A036036 for the reference and a C. F. Hindenburg link.
The present triangle is the partial sum triangle of the triangle t(n,k) = 0 if 0 <= n -1 < k , t(n,0) = 1, n >= 1 and t(n,k) = A008284(n,k) if n-1 >= k >= 1. This triangle with offset [1,0] for [n,k] is 1; 1,1; 1,1,1; 1,1,2,1; 1,1,2,2,1; 1,1,3,3,2,1;... (erase the diagonal of A008284 and add instead a column k=0 with only 1's). See the example section.
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LINKS
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FORMULA
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T(n,m) = sum(p(n,k),k=0..m-1) if n >= m >= 1, otherwise 0, with p(n,0) :=1 and p(n,k) = A008284(n,k) for k=1,2,...,n-1.
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EXAMPLE
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T(n,m) starts with:
n\m 1 2 3 4 5 6 7 8 9 10 11 12...
1 1
2 1 2
3 1 2 3
4 1 2 4 5
5 1 2 4 6 7
6 1 2 5 8 10 11
7 1 2 5 9 12 14 15
8 1 2 6 11 16 19 21 22
9 1 2 6 13 19 24 27 29 30
10 1 2 7 15 24 31 36 39 41 42
11 1 2 7 17 28 38 45 50 53 55 56
12 1 2 8 20 35 48 59 66 71 74 76 77
...
T(6,4) = 8 because the 11=T(6,6) partitions for n=6 are, in A-St order: [6]; [1,5],[2,4],[3,3]; [1^2,4],[1,2,3],[2^3]; [1^3,3],[1^2,2^2]; [1^4,2]; [1^6] and the first partition with 4 parts, appears at position 8.
This triangle is obtained as partial sum triangle from the triangle t(n,k) (see the comment section) which starts with:
n\m 0 1 2 3 4 5 6 7 8 9 10 11 ...
1 1
2 1 1
3 1 1 1
4 1 1 2 1
5 1 1 2 2 1
6 1 1 3 3 2 1
7 1 1 3 4 3 2 1
8 1 1 4 5 5 3 2 1
9 1 1 4 7 6 5 3 2 1
10 1 1 5 8 9 7 5 3 2 1
11 1 1 5 10 11 10 7 5 3 2 1
12 1 1 6 12 15 13 11 7 5 3 2 1
...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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