A286000 A table of partitions into consecutive parts (see Comments lines for definition).

1, 2, 3, 2, 4, 1, 5, 3, 6, 2, 3, 7, 4, 2, 8, 3, 1, 9, 5, 4, 10, 4, 3, 4, 11, 6, 2, 3, 12, 5, 5, 2, 13, 7, 4, 1, 14, 6, 3, 5, 15, 8, 6, 4, 5, 16, 7, 5, 3, 4, 17, 9, 4, 2, 3, 18, 8, 7, 6, 2, 19, 10, 6, 5, 1, 20, 9, 5, 4, 6, 21, 11, 8, 3, 5, 6, 22, 10, 7, 7, 4, 5, 23, 12, 6, 6, 3, 4, 24, 11, 9, 5, 2, 3, 25, 13, 8, 4, 7, 2

Offset: 1

Author

Omar E. Pol, Apr 30 2017

Keywords

• nonn
• tabf

Comments

This is a triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists successive blocks of k consecutive terms in decreasing order, where the m-th block starts with k + m - 1, m>=1, and the first element of column k is in the row k*(k+1)/2.

The partitions of n into consecutive parts are represented from the row n up to row A288529(n) as maximum, exclusively in the columns where the blocks begin.

More precisely, the partition of n into exactly k consecutive parts (if such partition exists) is represented in the column k from the row n up to row n + k - 1 (see examples).

A288772(n) is the minimum number of rows that are required to represent in this table the partitions of all positive integers <= n into consecutive parts.

A288773(n) is the largest of all positive integers whose partitions into consecutive parts can be totally represented in the first n rows of this table.

A288774(n) is the largest positive integers whose partitions into consecutive parts can be totally represented in the first n rows of this table.

Theorem: the smallest part of the partition of n into exactly k consecutive parts (if such partition exists) equals the number of positive integers <= n having a partition into exactly k consecutive parts.

Examples

			Table de partitions into consecutive parts (first 28 rows):
1;
2;
3,   2;
4,   1;
5,   3;
6,   2,  3;
7,   4,  2;
8,   3,  1;
9,   5,  4;
10,  4,  3,  4;
11,  6,  2,  3;
12,  5,  5,  2;
13,  7,  4,  1;
14,  6,  3,  5;
15,  8,  6,  4,  5;
16,  7,  5,  3,  4;
17,  9,  4,  2,  3;
18,  8,  7,  6,  2;
19, 10,  6,  5,  1;
20,  9,  5,  4,  6;
21, 11,  8,  3,  5,  6;
22, 10,  7,  7,  4,  5;
23, 12,  6,  6,  3,  4;
24, 11,  9,  5,  2,  3;
25, 13,  8,  4,  7,  2;
26, 12,  7,  8,  6,  1;
27, 14, 10,  7,  5,  7;
28, 13,  9,  6,  4,  6,  7;
...
Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts:
.   ------------------------------------------------------------------------
Fig:   A     B       C         D          E            F             G
.   ------------------------------------------------------------------------
. n:   1     2       3         4          5            6             7
Row ------------------------------------------------------------------------
1   | [1];|  1; |  1;     |  1;    |  1;        |  1;         |  1;        |
2   |     | [2];|  2;     |  2;    |  2;        |  2;         |  2;        |
3   |     |     | [3],[2];|  3;  2;|  3,  2;    |  3,  2;     |  3,  2;    |
4   |     |     |  4 ,[1];| [4], 1;|  4,  1;    |  4,  1;     |  4,  1;    |
5   |     |     |         |        | [5],[3];   |  5,  3;     |  5,  3;    |
6   |     |     |         |        |  6, [2], 3;| [6], 2, [3];|  6,  2,  3;|
7   |     |     |         |        |            |  7,  4, [2];| [7],[4], 2;|
8   |     |     |         |        |            |  8,  3, [1];|  8, [3], 1;|
.   ------------------------------------------------------------------------
Figure F: for n = 6 the partitions of 6 into consecutive parts are [6] and [3, 2, 1]. These partitions have 1 and 3 consecutive parts respectively. On the other hand  we can find the mentioned partitions in the columns 1 and 3 of this table, starting at the row 6.
.
Figures H..K show the location (in the columns of the table) of the partitions of 8..11 (respectively) into consecutive parts:
.    --------------------------------------------------------------------
Fig:        H             I                  J                 K
.    --------------------------------------------------------------------
. n:        8             9                  10                11
Row  --------------------------------------------------------------------
1    |  1;        |  1;            |   1;             |   1;            |
1    |  2;        |  2;            |   2;             |   2;            |
3    |  3,  2;    |  3,  2;        |   3,  2;         |   3,  2;        |
4    |  4,  1;    |  4,  1;        |   4,  1;         |   4,  1;        |
5    |  5,  3;    |  5,  3;        |   5,  3;         |   5,  3;        |
6    |  6,  2,  3;|  6,  2,  3;    |   6,  2,  3;     |   6,  2,  3;    |
7    |  7,  4,  2;|  7,  4,  2;    |   7,  4,  2;     |   7,  4,  2;    |
8    | [8], 3,  1;|  8,  3,  1;    |   8,  3,  1;     |   8,  3,  1;    |
9    |            | [9],[5],[4];   |   9,  5,  4;     |   9,  5,  4;    |
10   |            | 10, [4],[3], 4;| [10], 4,  3, [4];|  10,  4,  3;  4;|
11   |            | 11,  6, [2], 3;|  11,  6,  2; [3];| [11],[6], 2,  3;|
12   |            |                |  12,  5,  5, [2];|  12, [5], 5,  2;|
13   |            |                |  13,  7,  4, [1];|  13,  7,  4,  1;|
.    --------------------------------------------------------------------
Figure J: For n = 10 the partitions of 10 into consecutive parts are [10] and [4, 3, 2, 1]. These partitions have 1 and 4 consecutive parts respectively. On the other hand  we can find the mentioned partitions in the columns 1 and 4 of this table, starting at the row 10.
Illustration of initial terms arranged into the diagram of the triangle A237591:
.                                                           _
.                                                         _|1|
.                                                       _|2 _|
.                                                     _|3  |2|
.                                                   _|4   _|1|
.                                                 _|5    |3 _|
.                                               _|6     _|2|3|
.                                             _|7      |4  |2|
.                                           _|8       _|3 _|1|
.                                         _|9        |5  |4 _|
.                                       _|10        _|4  |3|4|
.                                     _|11         |6   _|2|3|
.                                   _|12          _|5  |5  |2|
.                                 _|13           |7    |4 _|1|
.                               _|14            _|6   _|3|5 _|
.                             _|15             |8    |6  |4|5|
.                           _|16              _|7    |5  |3|4|
.                         _|17               |9     _|4 _|2|3|
.                       _|18                _|8    |7  |6  |2|
.                     _|19                 |10     |6  |5 _|1|
.                   _|20                  _|9     _|5  |4|6 _|
.                 _|21                   |11     |8   _|3|5|6|
.               _|22                    _|10     |7  |7  |4|5|
.             _|23                     |12      _|6  |6  |3|4|
.           _|24                      _|11     |9    |5 _|2|3|
.         _|25                       |13       |8   _|4|7  |2|
.       _|26                        _|12      _|7  |8  |6 _|1|
.     _|27                         |14       |10   |7  |5|7 _|
.    |28                           |13       |9    |6  |4|6|7|
...
The number of horizontal line segments in the n-th row of the diagram equals A001227(n), the number of partitions of n into consecutive parts.
		

Crossrefs

Row n has length A003056(n).

The first element of column k is in row A000217(k).

For another version see A286001.

Cf. A001227, A109814, A196020, A204217, A235791, A236104, A237048, A237591, A237593, A245092, A285914, A286013, A288529, A288772, A288773, A288774.