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%I A286001 #59 Aug 02 2022 13:01:25 %S A286001 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, %T A286001 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, %U A286001 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 %N A286001 A table of partitions into consecutive parts (see Comments lines for definition). %C A286001 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. %C A286001 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. %C A286001 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 A286001 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 A286001 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 A286001 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 A286001 For a theorem related to this table see A286000. %e A286001 Triangle begins: %e A286001 1; %e A286001 2; %e A286001 3, 1; %e A286001 4, 2; %e A286001 5, 2; %e A286001 6, 3, 1; %e A286001 7, 3, 2; %e A286001 8, 4, 3; %e A286001 9, 4, 2; %e A286001 10, 5, 3, 1; %e A286001 11, 5, 4, 2; %e A286001 12, 6, 3, 3; %e A286001 13, 6, 4, 4; %e A286001 14, 7, 5, 2; %e A286001 15, 7, 4, 3, 1; %e A286001 16, 8, 5, 4, 2; %e A286001 17, 8, 6, 5, 3; %e A286001 18, 9, 5, 3, 4; %e A286001 19, 9, 6, 4, 5; %e A286001 20, 10, 7, 5, 2; %e A286001 21, 10, 6, 6, 3, 1; %e A286001 22, 11, 7, 4, 4, 2; %e A286001 23, 11, 8, 5, 5, 3; %e A286001 24, 12, 7, 6, 6, 4; %e A286001 25, 12, 8, 7, 3, 5; %e A286001 26, 13, 9, 5, 4, 6; %e A286001 27, 13, 8, 6, 5, 2; %e A286001 28, 14, 9, 7, 6, 3, 1; %e A286001 ... %e A286001 Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts: %e A286001 . ------------------------------------------------------------------------ %e A286001 Fig: A B C D E F G %e A286001 . ------------------------------------------------------------------------ %e A286001 . n: 1 2 3 4 5 6 7 %e A286001 Row ------------------------------------------------------------------------ %e A286001 1 | [1];| 1; | 1; | 1; | 1; | 1; | 1; | %e A286001 2 | | [2];| 2; | 2; | 2; | 2; | 2; | %e A286001 3 | | | [3],[1];| 3, 1;| 3, 1; | 3, 1; | 3, 1; | %e A286001 4 | | | 4 ,[2];| [4], 2;| 4, 2; | 4, 2; | 4, 2; | %e A286001 5 | | | | | [5],[2]; | 5, 2; | 5, 2; | %e A286001 6 | | | | | 6, [3], 3;| [6], 3, [1];| 6, 3, 1;| %e A286001 7 | | | | | | 7, 3, [2];| [7],[3], 2;| %e A286001 8 | | | | | | 8, 4, [3];| 8, [4], 3;| %e A286001 . ------------------------------------------------------------------------ %e A286001 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. %e A286001 . %e A286001 Figures H..K show the location (in the columns of the table) of the partitions of 8..11 (respectively) into consecutive parts: %e A286001 . -------------------------------------------------------------------- %e A286001 Fig: H I J K %e A286001 . -------------------------------------------------------------------- %e A286001 . n: 8 9 10 11 %e A286001 Row -------------------------------------------------------------------- %e A286001 1 | 1; | 1; | 1; | 1; | %e A286001 1 | 2; | 2; | 2; | 2; | %e A286001 3 | 3, 1; | 3, 1; | 3, 1; | 3, 1; | %e A286001 4 | 4, 2; | 4, 2; | 4, 2; | 4, 2; | %e A286001 5 | 5, 2; | 5, 2; | 5, 2; | 5, 2; | %e A286001 6 | 6, 3, 3;| 6, 3, 1; | 6, 3, 1; | 6, 3, 1; | %e A286001 7 | 7, 3, 2;| 7, 3, 2; | 7, 3, 2; | 7, 3, 2; | %e A286001 8 | [8], 4, 1;| 8, 4, 3; | 8, 4, 3; | 8, 4, 3; | %e A286001 9 | | [9],[4],[2]; | 9, 4, 2; | 9, 4, 2; | %e A286001 10 | | 10, [5],[3], 1;| [10], 5, 3, [1];| 10, 5, 3, 1;| %e A286001 11 | | 11, 5, [4], 2;| 11, 5, 4, [2];| [11],[5], 4, 2;| %e A286001 12 | | | 12, 6, 3, [3];| 12, [6], 3, 3;| %e A286001 13 | | | 13, 6, 4, [4];| 13, 6, 4, 4;| %e A286001 . -------------------------------------------------------------------- %e A286001 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. %e A286001 . %e A286001 Illustration of initial terms arranged into the diagram of the triangle A237591: %e A286001 . _ %e A286001 . _|1| %e A286001 . _|2 _| %e A286001 . _|3 |1| %e A286001 . _|4 _|2| %e A286001 . _|5 |2 _| %e A286001 . _|6 _|3|1| %e A286001 . _|7 |3 |2| %e A286001 . _|8 _|4 _|3| %e A286001 . _|9 |4 |2 _| %e A286001 . _|10 _|5 |3|1| %e A286001 . _|11 |5 _|4|2| %e A286001 . _|12 _|6 |3 |3| %e A286001 . _|13 |6 |4 _|4| %e A286001 . _|14 _|7 _|5|2 _| %e A286001 . _|15 |7 |4 |3|1| %e A286001 . _|16 _|8 |5 |4|2| %e A286001 . _|17 |8 _|6 _|5|3| %e A286001 . _|18 _|9 |5 |3 |4| %e A286001 . _|19 |9 |6 |4 _|5| %e A286001 . _|20 _|10 _|7 |5|2 _| %e A286001 . _|21 |10 |6 _|6|3|1| %e A286001 . _|22 _|11 |7 |4 |4|2| %e A286001 . _|23 |11 _|8 |5 |5|3| %e A286001 . _|24 _|12 |7 |6 _|6|4| %e A286001 . _|25 |12 |8 _|7|3 |5| %e A286001 . _|26 _|13 _|9 |5 |4 _|6| %e A286001 . _|27 |13 |8 |6 |5|2 _| %e A286001 . |28 |14 |9 |7 |6|3|1| %e A286001 ... %e A286001 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. %e A286001 . %e A286001 From _Omar E. Pol_, Dec 15 2020: (Start) %e A286001 The connection (described step by step) between the triangle of A299765 and the above geometric diagram is as follows: %e A286001 . %e A286001 [1]; [1]; %e A286001 [2]; [2]; %e A286001 [3], [2, 1]; [3], [2, 1]; %e A286001 [4]; [4]; %e A286001 [5], [3, 2]; [5], [3, 2]; %e A286001 [6], [3, 2, 1]; [6], [3, 2, 1]; %e A286001 [7], [4, 3]; [7], [4, 3]; %e A286001 [8]; [8]; %e A286001 [9], [5, 4], [4, 3, 2]; [9], [5, 4], [4, 3, 2]; %e A286001 . %e A286001 Figure 1. Figure 2. %e A286001 . %e A286001 We start with the irregular Then we write the same triangle %e A286001 triangle of A299765 in which but ordered in columns where the %e A286001 row n lists the partitions column k lists the partitions of %e A286001 of n into consecutive parts. n into k consecutive parts. %e A286001 . %e A286001 . _ _ %e A286001 1| |1 %e A286001 _ _ %e A286001 2| |2 %e A286001 _ _ _ _ _ %e A286001 3| 2,1| |3 |1 %e A286001 _ _ |2 %e A286001 4| |4 %e A286001 _ _ _ _ _ %e A286001 5| 3,2| |5 |2 %e A286001 _ _ _ _ _ |3 _ %e A286001 6| 3,2,1| |6 |1 %e A286001 _ _ _ _ _ |2 %e A286001 7| 4,3| |7 |3 |3 %e A286001 _ _ |4 %e A286001 8| |8 %e A286001 _ _ _ _ _ _ _ _ _ %e A286001 9| 5,4| 4,3,2| |9 |4 |2 %e A286001 |5 |3 %e A286001 |4 %e A286001 . %e A286001 Figure 3. Figure 4. %e A286001 . %e A286001 Then we draw to the right of Then we rotate each sub-diagram %e A286001 each partition a vertical 90 degrees counterclockwise. %e A286001 toothpick and above each part Every horizontal toothpick represents %e A286001 we draw a horizontal toothpick. the existence of that partition. %e A286001 . The number of vertical toothpicks %e A286001 . equals the number of parts. %e A286001 . %e A286001 . _ _ %e A286001 _|1 _|1 %e A286001 _|2 _ _|2 _ %e A286001 _|3 |1 _|3 |1 %e A286001 _|4 _|2 _|4 _|2 %e A286001 _|5 |2 _ _|5 |2 _ %e A286001 _|6 _|3|1 _|6 _|3|1 %e A286001 _|7 |3 |2 _|7 |3 |2 %e A286001 _|8 _|4 _|3 _|8 _|4 _|3 %e A286001 |9 |4 |2 |9 |4 |2 %e A286001 |5 |3 %e A286001 |4 %e A286001 . %e A286001 Figure 5. Figure 6. %e A286001 . %e A286001 Then we join the sub-diagrams Finally we erase the parts that %e A286001 forming staircases (or zig-zag are beyond a certain level (in %e A286001 paths) that represent the this case beyond the 9th level) %e A286001 partitions that have the same to make the diagram more standard. %e A286001 number of parts. %e A286001 . %e A286001 The numbers in the k-th staircase (from left to right) are the elements of the k-th column of the triangular array. %e A286001 Note that this diagram is essentially the same diagram used to represent the triangles A237048, A235791, A237591, and other related sequences such as A001227, A060831 and A204217. %e A286001 There is an infinite family of this kind of triangles, which are related to polygonal numbers and partitions into consecutive parts that differ by d. For more information see the theorems in A285914 and A303300. %e A286001 Note that if we take two images of the diagram mirroring each other, with the y-axis in the middle of them, then a new diagram is formed, which is symmetric and represents the sequence A237593 as an isosceles triangle. Then if we fold each level (or row) of that isosceles triangle we essentially obtain the structure of the pyramid described in A245092 whose terraces at the n-th level have a total area equal to sigma(n) = A000203(n). (End) %Y A286001 Another version of A286000. %Y A286001 Tables of the same family where the consecutive parts differ by d are A010766 (d=0), this sequence (d=1), A332266 (d=2), A334945 (d=3), A334618(d=4). %Y A286001 Cf. A000217, A001227, A003056, A109814, A196020, A204217, A211343, A235791, A236104, A237048, A237591, A237593, A245092, A262626, A280850, A280851, A285914, A286013, A288529, A288772, A288773, A288774, A296508, A299765, A303300. %K A286001 nonn,tabl %O A286001 1,2 %A A286001 _Omar E. Pol_, Apr 30 2017