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A228716 Triangle read by rows in which row n lists the rows (including 0's) of the n-th section of the set of partitions (in colexicographic order) of any integer >= n.

%I A228716 #27 Mar 12 2015 20:53:28
%S A228716 1,0,1,2,0,0,1,0,1,3,0,0,0,1,0,0,1,0,1,2,2,4,0,0,0,0,1,0,0,0,1,0,0,1,
%T A228716 0,0,1,0,1,3,2,5,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,0,1,0,0,1,0,0,1,0,
%U A228716 1,2,2,2,4,2,3,3,6,0,0,0,0,0,0,1,0,0
%N A228716 Triangle read by rows in which row n lists the rows (including 0's) of the n-th section of the set of partitions (in colexicographic order) of any integer >= n.
%C A228716 In other words, row n lists the rows of the last section of the set of partitions (in colexicographic order) of n.
%C A228716 Row lengths is A006128.
%C A228716 The number of zeros in row n is A006128(n-1).
%C A228716 Rows sums give A138879.
%C A228716 For more properties of the sections of the set of partitions of a positive integer see example.
%C A228716 Positive terms give A230440. - _Omar E. Pol_, Oct 25 2013
%e A228716 Illustration of the 15 rows of the 7th section (including zeros) of the set of partitions of any integer >= 7 (hence this is also the last section of the set of partitions of 7). Note that the sum of the k-th column is equal to the number of parts >= k, therefore the first differences of the column sums give the number of occurrences of parts k in the section. The same for all sections of all positive integers, see below:
%e A228716 -----------------------------
%e A228716 Column: 1  2  3  4  5  6  7
%e A228716 -----------------------------
%e A228716 Row |
%e A228716 1   |   0, 0, 0, 0, 0, 0, 1;
%e A228716 2   |   0, 0, 0, 0, 0, 1;
%e A228716 3   |   0, 0, 0, 0, 1;
%e A228716 4   |   0, 0, 0, 0, 1;
%e A228716 5   |   0, 0, 0, 1;
%e A228716 6   |   0, 0, 0, 1;
%e A228716 7   |   0, 0, 1;
%e A228716 8   |   0, 0, 0, 1;
%e A228716 9   |   0, 0, 1;
%e A228716 10  |   0, 0, 1;
%e A228716 11  |   0, 1;
%e A228716 12  |   3, 2, 2;
%e A228716 13  |   5, 2;
%e A228716 14  |   4, 3;
%e A228716 15  |   7;
%e A228716 -----------------------------
%e A228716 Sums:  19, 8, 5, 3, 2, 1, 1 -> Row 7 of triangle A207031.
%e A228716 .       | /| /| /| /| /| /|
%e A228716 .       |/ |/ |/ |/ |/ |/ |
%e A228716 F.Dif: 11, 3, 2, 1, 1, 0, 1 -> Row 7 of triangle A182703.
%e A228716 .
%e A228716 Triangle begins:
%e A228716 [1];
%e A228716 [0,1],[2];
%e A228716 [0,0,1],[0,1],[3];
%e A228716 [0,0,0,1],[0,0,1],[0,1],[2,2],[4];
%e A228716 [0,0,0,0,1],[0,0,0,1],[0,0,1],[0,0,1],[0,1],[3,2],[5];
%e A228716 [0,0,0,0,0,1],[0,0,0,0,1],[0,0,0,1],[0,0,0,1],[0,0,1],[0,0,1],[0,1],[2,2,2],[4,2],[3,3],[6];
%e A228716 [0,0,0,0,0,0,1],[0,0,0,0,0,1],[0,0,0,0,1],[0,0,0,0,1],[0,0,0,1],[0,0,0,1],[0,0,1],[0,0,0,1],[0,0,1],[0,0,1],[0,1],[3,2,2],[5,2],[4,3],[7];
%Y A228716 Cf. A000041, A006128, A135010, A138121, A138879, A182703, A187219, A207031, A211992.
%K A228716 nonn,tabf
%O A228716 1,4
%A A228716 _Omar E. Pol_, Sep 02 2013