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