A228716 - OEIS

%I #27 Mar 12 2015 20:53:28

%S 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 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 1,2,2,2,4,2,3,3,6,0,0,0,0,0,0,1,0,0

%N 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 In other words, row n lists the rows of the last section of the set of partitions (in colexicographic order) of n.

%C Row lengths is A006128.

%C The number of zeros in row n is A006128(n-1).

%C Rows sums give A138879.

%C For more properties of the sections of the set of partitions of a positive integer see example.

%C Positive terms give A230440. - _Omar E. Pol_, Oct 25 2013

%e 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 -----------------------------

%e Column: 1  2  3  4  5  6  7

%e -----------------------------

%e Row |

%e 1   |   0, 0, 0, 0, 0, 0, 1;

%e 2   |   0, 0, 0, 0, 0, 1;

%e 3   |   0, 0, 0, 0, 1;

%e 4   |   0, 0, 0, 0, 1;

%e 5   |   0, 0, 0, 1;

%e 6   |   0, 0, 0, 1;

%e 7   |   0, 0, 1;

%e 8   |   0, 0, 0, 1;

%e 9   |   0, 0, 1;

%e 10  |   0, 0, 1;

%e 11  |   0, 1;

%e 12  |   3, 2, 2;

%e 13  |   5, 2;

%e 14  |   4, 3;

%e 15  |   7;

%e -----------------------------

%e Sums:  19, 8, 5, 3, 2, 1, 1 -> Row 7 of triangle A207031.

%e .       | /| /| /| /| /| /|

%e .       |/ |/ |/ |/ |/ |/ |

%e F.Dif: 11, 3, 2, 1, 1, 0, 1 -> Row 7 of triangle A182703.

%e .

%e Triangle begins:

%e [1];

%e [0,1],[2];

%e [0,0,1],[0,1],[3];

%e [0,0,0,1],[0,0,1],[0,1],[2,2],[4];

%e [0,0,0,0,1],[0,0,0,1],[0,0,1],[0,0,1],[0,1],[3,2],[5];

%e [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 [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 Cf. A000041, A006128, A135010, A138121, A138879, A182703, A187219, A207031, A211992.

%K nonn,tabf

%O 1,4

%A _Omar E. Pol_, Sep 02 2013