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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).

%I A185370 #41 Mar 12 2015 21:06:21
%S A185370 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,
%T A185370 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,
%U A185370 1,0,0,0,0,1,0,0,0,1,15,4,1,1,0,0,0,1
%N 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).
%C A185370 For the definition of "region of the set of partitions of j" see A206437.
%C A185370 T(n,k) is the number of occurrences of k in the n-th region of the shell model of partitions (see A135010).
%C A185370 T(n,k) is also the number of occurrences of k in the n-th row of triangles A186114, A193870, A206437 (and possibly more).
%C A185370 If the length of row n is a record then the length of row n is j and also A000041(j) = n.
%C A185370 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).
%H A185370 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpar02.jpg">Illustration of the seven regions of 5</a>
%e A185370 First seven regions of any integer >= 5 are
%e A185370 [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].
%e A185370 -----------------------------------------
%e A185370 n    j  m    k : 1  2  3  4  5  6  7  8
%e A185370 -----------------------------------------
%e A185370 1    1  1        1;
%e A185370 2    2  1        1, 1;
%e A185370 3    3  1        2, 0, 1;
%e A185370 4    4  1        0, 1;
%e A185370 5    4  2        3, 1, 0, 1;
%e A185370 6    5  1        0, 0, 1;
%e A185370 7    5  2        5, 1, 0, 0, 1;
%e A185370 8    6  1        0, 1;
%e A185370 9    6  2        0, 1, 0, 1;
%e A185370 10   6  3        0, 0, 1;
%e A185370 11   6  4        7, 2, 1, 0, 0, 1;
%e A185370 12   7  1        0, 0, 1;
%e A185370 13   7  2        0, 1, 0, 0, 1;
%e A185370 14   7  3        0, 0, 0, 1;
%e A185370 15   7  4       11, 2, 1, 0, 0, 0, 1;
%e A185370 16   8  1        0, 1;
%e A185370 17   8  2        0, 1, 0, 1;
%e A185370 18   8  3        0, 0, 1;
%e A185370 19   8  4        0, 2, 1, 0, 0, 1;
%e A185370 20   8  5        0, 0, 0, 0, 1;
%e A185370 21   8  6        0, 0, 0, 1;
%e A185370 22   8  7       15, 4, 1, 1, 0, 0, 0, 1;
%Y A185370 Row n has length A141285(n). Row sums give A194446. Positive terms of column 1 give A000041.
%Y A185370 Cf. A006128, A066633, A135010, A182703, A186114, A187219, A193870, A206437.
%K A185370 nonn,tabf
%O A185370 1,4
%A A185370 _Omar E. Pol_, Jan 25 2013