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A286001 A table of partitions into consecutive parts (see Comments lines for definition). 16
1, 2, 3, 1, 4, 2, 5, 2, 6, 3, 1, 7, 3, 2, 8, 4, 3, 9, 4, 2, 10, 5, 3, 1, 11, 5, 4, 2, 12, 6, 3, 3, 13, 6, 4, 4, 14, 7, 5, 2, 15, 7, 4, 3, 1, 16, 8, 5, 4, 2, 17, 8, 6, 5, 3, 18, 9, 5, 3, 4, 19, 9, 6, 4, 5, 20, 10, 7, 5, 2, 21, 10, 6, 6, 3, 1, 22, 11, 7, 4, 4, 2, 23, 11, 8, 5, 5, 3, 24, 12, 7, 6, 6, 4, 25, 12, 8, 7, 3, 5 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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, where the m-th block starts with m, m>=1, and the first element of column k is in 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, but in increasing order, 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.

For a theorem related to this table see A286000.

LINKS

Table of n, a(n) for n=1..100.

EXAMPLE

Triangle begins:

1;

2;

3,   1;

4,   2;

5,   2;

6,   3,  1;

7,   3,  2;

8,   4,  3;

9,   4,  2;

10,  5,  3,  1;

11,  5,  4,  2;

12,  6,  3,  3;

13,  6,  4,  4;

14,  7,  5,  2;

15,  7,  4,  3,  1;

16,  8,  5,  4,  2;

17,  8,  6,  5,  3;

18,  9,  5,  3,  4;

19,  9,  6,  4,  5;

20, 10,  7,  5,  2;

21, 10,  6,  6,  3,  1;

22, 11,  7,  4,  4,  2;

23, 11,  8,  5,  5,  3;

24, 12,  7,  6,  6,  4;

25, 12,  8,  7,  3,  5;

26, 13,  9,  5,  4,  6;

27, 13,  8,  6,  5,  2;

28, 14,  9,  7,  6,  3,  1;

...

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],[1];|  3;  1;|  3,  1;    |  3,  1;     |  3,  1;    |

4   |     |     |  4 ,[2];| [4], 2;|  4,  2;    |  4,  2;     |  4,  2;    |

5   |     |     |         |        | [5],[2];   |  5,  2;     |  5,  2;    |

6   |     |     |         |        |  6, [3], 3;| [6], 3, [1];|  6,  3,  1;|

7   |     |     |         |        |            |  7,  3, [2];| [7],[3], 2;|

8   |     |     |         |        |            |  8,  4, [3];|  8, [4], 3;|

.   ------------------------------------------------------------------------

Figure F: for n = 6 the partitions of 6 into consecutive parts (but with the parts in increasing order) are [6] and [1, 2, 3]. 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,  1;    |  3,  1;        |   3,  1;         |   3,  1;        |

4    |  4,  2;    |  4,  2;        |   4,  2;         |   4,  2;        |

5    |  5,  2;    |  5,  2;        |   5,  2;         |   5,  2;        |

6    |  6,  3,  3;|  6,  3,  1;    |   6,  3,  1;     |   6,  3,  1;    |

7    |  7,  3,  2;|  7,  3,  2;    |   7,  3,  2;     |   7,  3,  2;    |

8    | [8], 4,  1;|  8,  4,  3;    |   8,  4,  3;     |   8,  4,  3;    |

9    |            | [9],[4],[2];   |   9,  4,  2;     |   9,  4,  2;    |

10   |            | 10, [5],[3], 1;| [10], 5,  3, [1];|  10,  5,  3;  1;|

11   |            | 11,  5, [4], 2;|  11,  5,  4; [2];| [11],[5], 4,  2;|

12   |            |                |  12,  6,  3, [3];|  12, [6], 3,  3;|

13   |            |                |  13,  6,  4, [4];|  13,  6,  4,  4;|

.    --------------------------------------------------------------------

Figure J: For n = 10 the partitions of 10 into consecutive parts (but with the parts in increasing order) are [10] and [1, 2, 3, 4]. 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  |1|

.                                                   _|4   _|2|

.                                                 _|5    |2 _|

.                                               _|6     _|3|1|

.                                             _|7      |3  |2|

.                                           _|8       _|4 _|3|

.                                         _|9        |4  |2 _|

.                                       _|10        _|5  |3|1|

.                                     _|11         |5   _|4|2|

.                                   _|12          _|6  |3  |3|

.                                 _|13           |6    |4 _|4|

.                               _|14            _|7   _|5|2 _|

.                             _|15             |7    |4  |3|1|

.                           _|16              _|8    |5  |4|2|

.                         _|17               |8     _|6 _|5|3|

.                       _|18                _|9    |5  |3  |4|

.                     _|19                 |9      |6  |4 _|5|

.                   _|20                  _|10    _|7  |5|2 _|

.                 _|21                   |10     |6   _|6|3|1|

.               _|22                    _|11     |7  |4  |4|2|

.             _|23                     |11      _|8  |5  |5|3|

.           _|24                      _|12     |7    |6 _|6|4|

.         _|25                       |12       |8   _|7|3  |5|

.       _|26                        _|13      _|9  |5  |4 _|6|

.     _|27                         |13       |8    |6  |5|2 _|

.    |28                           |14       |9    |7  |6|3|1|

...

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

Another version of A286000.

Cf. A000217, A001227, A003056, A109814, A196020, A204217, A211343, A235791, A236104, A237048, A237591, A237593, A245092, A285914, A286013, A288529, A288772, A288773, A288774.

Sequence in context: A152547 A083906 A160541 * A304106 A022446 A122196

Adjacent sequences:  A285998 A285999 A286000 * A286002 A286003 A286004

KEYWORD

nonn,tabl

AUTHOR

Omar E. Pol, Apr 30 2017

STATUS

approved

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Last modified April 21 07:53 EDT 2019. Contains 322327 sequences. (Running on oeis4.)