A214314 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.

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

Offset: 1

Author

Wolfdieter Lang, Jul 24 2012

Keywords

• nonn
• tabl

Comments

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.

Examples

			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
...
		

Crossrefs

Cf. A008284.

Formula

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.