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A table of partitions into consecutive parts (see Comments lines for definition).
37

%I #66 Oct 21 2017 21:05:29

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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.

%C 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).

%C 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.

%C 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.

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

%C 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.

%e Table de partitions into consecutive parts (first 28 rows):

%e 1;

%e 2;

%e 3, 2;

%e 4, 1;

%e 5, 3;

%e 6, 2, 3;

%e 7, 4, 2;

%e 8, 3, 1;

%e 9, 5, 4;

%e 10, 4, 3, 4;

%e 11, 6, 2, 3;

%e 12, 5, 5, 2;

%e 13, 7, 4, 1;

%e 14, 6, 3, 5;

%e 15, 8, 6, 4, 5;

%e 16, 7, 5, 3, 4;

%e 17, 9, 4, 2, 3;

%e 18, 8, 7, 6, 2;

%e 19, 10, 6, 5, 1;

%e 20, 9, 5, 4, 6;

%e 21, 11, 8, 3, 5, 6;

%e 22, 10, 7, 7, 4, 5;

%e 23, 12, 6, 6, 3, 4;

%e 24, 11, 9, 5, 2, 3;

%e 25, 13, 8, 4, 7, 2;

%e 26, 12, 7, 8, 6, 1;

%e 27, 14, 10, 7, 5, 7;

%e 28, 13, 9, 6, 4, 6, 7;

%e ...

%e Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts:

%e . ------------------------------------------------------------------------

%e Fig: A B C D E F G

%e . ------------------------------------------------------------------------

%e . n: 1 2 3 4 5 6 7

%e Row ------------------------------------------------------------------------

%e 1 | [1];| 1; | 1; | 1; | 1; | 1; | 1; |

%e 2 | | [2];| 2; | 2; | 2; | 2; | 2; |

%e 3 | | | [3],[2];| 3; 2;| 3, 2; | 3, 2; | 3, 2; |

%e 4 | | | 4 ,[1];| [4], 1;| 4, 1; | 4, 1; | 4, 1; |

%e 5 | | | | | [5],[3]; | 5, 3; | 5, 3; |

%e 6 | | | | | 6, [2], 3;| [6], 2, [3];| 6, 2, 3;|

%e 7 | | | | | | 7, 4, [2];| [7],[4], 2;|

%e 8 | | | | | | 8, 3, [1];| 8, [3], 1;|

%e . ------------------------------------------------------------------------

%e 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.

%e .

%e Figures H..K show the location (in the columns of the table) of the partitions of 8..11 (respectively) into consecutive parts:

%e . --------------------------------------------------------------------

%e Fig: H I J K

%e . --------------------------------------------------------------------

%e . n: 8 9 10 11

%e Row --------------------------------------------------------------------

%e 1 | 1; | 1; | 1; | 1; |

%e 1 | 2; | 2; | 2; | 2; |

%e 3 | 3, 2; | 3, 2; | 3, 2; | 3, 2; |

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

%e 5 | 5, 3; | 5, 3; | 5, 3; | 5, 3; |

%e 6 | 6, 2, 3;| 6, 2, 3; | 6, 2, 3; | 6, 2, 3; |

%e 7 | 7, 4, 2;| 7, 4, 2; | 7, 4, 2; | 7, 4, 2; |

%e 8 | [8], 3, 1;| 8, 3, 1; | 8, 3, 1; | 8, 3, 1; |

%e 9 | | [9],[5],[4]; | 9, 5, 4; | 9, 5, 4; |

%e 10 | | 10, [4],[3], 4;| [10], 4, 3, [4];| 10, 4, 3; 4;|

%e 11 | | 11, 6, [2], 3;| 11, 6, 2; [3];| [11],[6], 2, 3;|

%e 12 | | | 12, 5, 5, [2];| 12, [5], 5, 2;|

%e 13 | | | 13, 7, 4, [1];| 13, 7, 4, 1;|

%e . --------------------------------------------------------------------

%e 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.

%e Illustration of initial terms arranged into the diagram of the triangle A237591:

%e . _

%e . _|1|

%e . _|2 _|

%e . _|3 |2|

%e . _|4 _|1|

%e . _|5 |3 _|

%e . _|6 _|2|3|

%e . _|7 |4 |2|

%e . _|8 _|3 _|1|

%e . _|9 |5 |4 _|

%e . _|10 _|4 |3|4|

%e . _|11 |6 _|2|3|

%e . _|12 _|5 |5 |2|

%e . _|13 |7 |4 _|1|

%e . _|14 _|6 _|3|5 _|

%e . _|15 |8 |6 |4|5|

%e . _|16 _|7 |5 |3|4|

%e . _|17 |9 _|4 _|2|3|

%e . _|18 _|8 |7 |6 |2|

%e . _|19 |10 |6 |5 _|1|

%e . _|20 _|9 _|5 |4|6 _|

%e . _|21 |11 |8 _|3|5|6|

%e . _|22 _|10 |7 |7 |4|5|

%e . _|23 |12 _|6 |6 |3|4|

%e . _|24 _|11 |9 |5 _|2|3|

%e . _|25 |13 |8 _|4|7 |2|

%e . _|26 _|12 _|7 |8 |6 _|1|

%e . _|27 |14 |10 |7 |5|7 _|

%e . |28 |13 |9 |6 |4|6|7|

%e ...

%e 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.

%Y Row n has length A003056(n).

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

%Y For another version see A286001.

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

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Apr 30 2017