A185370 Triangle read by rows: T(n,k) is the number of occurrences of k in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

1, 1, 1, 2, 0, 1, 0, 1, 3, 1, 0, 1, 0, 0, 1, 5, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 7, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 11, 2, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 15, 4, 1, 1, 0, 0, 0, 1

Offset: 1

Author

Omar E. Pol, Jan 25 2013

Keywords

• nonn
• tabf

Comments

For the definition of "region of the set of partitions of j" see A206437.

T(n,k) is the number of occurrences of k in the n-th region of the shell model of partitions (see A135010).

T(n,k) is also the number of occurrences of k in the n-th row of triangles A186114, A193870, A206437 (and possibly more).

If the length of row n is a record then the length of row n is j and also A000041(j) = n.

If A000041(j) = n then the sum of the last A187219(j) elements of column k is A182703(j,k) and also the sum of all elements of column k is A066633(j,k).

Examples

			First seven regions of any integer >= 5 are
[1], [2,1], [3,1,1], [2], [4,2,1,1,1], [3], [5,2,1,1,1,1,1] (see illustrations, see also A206437). The 7th region contains five 1's, only one 2 and only one 5. There are no 3's. There are no 4's, so row 7 is [5, 1, 0, 0, 1].
-----------------------------------------
n    j  m    k : 1  2  3  4  5  6  7  8
-----------------------------------------
1    1  1        1;
2    2  1        1, 1;
3    3  1        2, 0, 1;
4    4  1        0, 1;
5    4  2        3, 1, 0, 1;
6    5  1        0, 0, 1;
7    5  2        5, 1, 0, 0, 1;
8    6  1        0, 1;
9    6  2        0, 1, 0, 1;
10   6  3        0, 0, 1;
11   6  4        7, 2, 1, 0, 0, 1;
12   7  1        0, 0, 1;
13   7  2        0, 1, 0, 0, 1;
14   7  3        0, 0, 0, 1;
15   7  4       11, 2, 1, 0, 0, 0, 1;
16   8  1        0, 1;
17   8  2        0, 1, 0, 1;
18   8  3        0, 0, 1;
19   8  4        0, 2, 1, 0, 0, 1;
20   8  5        0, 0, 0, 0, 1;
21   8  6        0, 0, 0, 1;
22   8  7       15, 4, 1, 1, 0, 0, 0, 1;
		

Links

• Omar E. Pol, Illustration of the seven regions of 5

Crossrefs

Row n has length A141285(n). Row sums give A194446. Positive terms of column 1 give A000041.

Cf. A006128, A066633, A135010, A182703, A186114, A187219, A193870, A206437.