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%I A286000 #66 Oct 21 2017 21:05:29 %S A286000 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 A286000 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 A286000 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 A286000 A table of partitions into consecutive parts (see Comments lines for definition). %C A286000 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 A286000 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 A286000 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 A286000 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 A286000 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 A286000 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 A286000 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 A286000 Table de partitions into consecutive parts (first 28 rows): %e A286000 1; %e A286000 2; %e A286000 3, 2; %e A286000 4, 1; %e A286000 5, 3; %e A286000 6, 2, 3; %e A286000 7, 4, 2; %e A286000 8, 3, 1; %e A286000 9, 5, 4; %e A286000 10, 4, 3, 4; %e A286000 11, 6, 2, 3; %e A286000 12, 5, 5, 2; %e A286000 13, 7, 4, 1; %e A286000 14, 6, 3, 5; %e A286000 15, 8, 6, 4, 5; %e A286000 16, 7, 5, 3, 4; %e A286000 17, 9, 4, 2, 3; %e A286000 18, 8, 7, 6, 2; %e A286000 19, 10, 6, 5, 1; %e A286000 20, 9, 5, 4, 6; %e A286000 21, 11, 8, 3, 5, 6; %e A286000 22, 10, 7, 7, 4, 5; %e A286000 23, 12, 6, 6, 3, 4; %e A286000 24, 11, 9, 5, 2, 3; %e A286000 25, 13, 8, 4, 7, 2; %e A286000 26, 12, 7, 8, 6, 1; %e A286000 27, 14, 10, 7, 5, 7; %e A286000 28, 13, 9, 6, 4, 6, 7; %e A286000 ... %e A286000 Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts: %e A286000 . ------------------------------------------------------------------------ %e A286000 Fig: A B C D E F G %e A286000 . ------------------------------------------------------------------------ %e A286000 . n: 1 2 3 4 5 6 7 %e A286000 Row ------------------------------------------------------------------------ %e A286000 1 | [1];| 1; | 1; | 1; | 1; | 1; | 1; | %e A286000 2 | | [2];| 2; | 2; | 2; | 2; | 2; | %e A286000 3 | | | [3],[2];| 3; 2;| 3, 2; | 3, 2; | 3, 2; | %e A286000 4 | | | 4 ,[1];| [4], 1;| 4, 1; | 4, 1; | 4, 1; | %e A286000 5 | | | | | [5],[3]; | 5, 3; | 5, 3; | %e A286000 6 | | | | | 6, [2], 3;| [6], 2, [3];| 6, 2, 3;| %e A286000 7 | | | | | | 7, 4, [2];| [7],[4], 2;| %e A286000 8 | | | | | | 8, 3, [1];| 8, [3], 1;| %e A286000 . ------------------------------------------------------------------------ %e A286000 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 A286000 . %e A286000 Figures H..K show the location (in the columns of the table) of the partitions of 8..11 (respectively) into consecutive parts: %e A286000 . -------------------------------------------------------------------- %e A286000 Fig: H I J K %e A286000 . -------------------------------------------------------------------- %e A286000 . n: 8 9 10 11 %e A286000 Row -------------------------------------------------------------------- %e A286000 1 | 1; | 1; | 1; | 1; | %e A286000 1 | 2; | 2; | 2; | 2; | %e A286000 3 | 3, 2; | 3, 2; | 3, 2; | 3, 2; | %e A286000 4 | 4, 1; | 4, 1; | 4, 1; | 4, 1; | %e A286000 5 | 5, 3; | 5, 3; | 5, 3; | 5, 3; | %e A286000 6 | 6, 2, 3;| 6, 2, 3; | 6, 2, 3; | 6, 2, 3; | %e A286000 7 | 7, 4, 2;| 7, 4, 2; | 7, 4, 2; | 7, 4, 2; | %e A286000 8 | [8], 3, 1;| 8, 3, 1; | 8, 3, 1; | 8, 3, 1; | %e A286000 9 | | [9],[5],[4]; | 9, 5, 4; | 9, 5, 4; | %e A286000 10 | | 10, [4],[3], 4;| [10], 4, 3, [4];| 10, 4, 3; 4;| %e A286000 11 | | 11, 6, [2], 3;| 11, 6, 2; [3];| [11],[6], 2, 3;| %e A286000 12 | | | 12, 5, 5, [2];| 12, [5], 5, 2;| %e A286000 13 | | | 13, 7, 4, [1];| 13, 7, 4, 1;| %e A286000 . -------------------------------------------------------------------- %e A286000 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 A286000 Illustration of initial terms arranged into the diagram of the triangle A237591: %e A286000 . _ %e A286000 . _|1| %e A286000 . _|2 _| %e A286000 . _|3 |2| %e A286000 . _|4 _|1| %e A286000 . _|5 |3 _| %e A286000 . _|6 _|2|3| %e A286000 . _|7 |4 |2| %e A286000 . _|8 _|3 _|1| %e A286000 . _|9 |5 |4 _| %e A286000 . _|10 _|4 |3|4| %e A286000 . _|11 |6 _|2|3| %e A286000 . _|12 _|5 |5 |2| %e A286000 . _|13 |7 |4 _|1| %e A286000 . _|14 _|6 _|3|5 _| %e A286000 . _|15 |8 |6 |4|5| %e A286000 . _|16 _|7 |5 |3|4| %e A286000 . _|17 |9 _|4 _|2|3| %e A286000 . _|18 _|8 |7 |6 |2| %e A286000 . _|19 |10 |6 |5 _|1| %e A286000 . _|20 _|9 _|5 |4|6 _| %e A286000 . _|21 |11 |8 _|3|5|6| %e A286000 . _|22 _|10 |7 |7 |4|5| %e A286000 . _|23 |12 _|6 |6 |3|4| %e A286000 . _|24 _|11 |9 |5 _|2|3| %e A286000 . _|25 |13 |8 _|4|7 |2| %e A286000 . _|26 _|12 _|7 |8 |6 _|1| %e A286000 . _|27 |14 |10 |7 |5|7 _| %e A286000 . |28 |13 |9 |6 |4|6|7| %e A286000 ... %e A286000 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 A286000 Row n has length A003056(n). %Y A286000 The first element of column k is in row A000217(k). %Y A286000 For another version see A286001. %Y A286000 Cf. A001227, A109814, A196020, A204217, A235791, A236104, A237048, A237591, A237593, A245092, A285914, A286013, A288529, A288772, A288773, A288774. %K A286000 nonn,tabf %O A286000 1,2 %A A286000 _Omar E. Pol_, Apr 30 2017