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A299765
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Irregular triangle read by rows, T(n,k), n >= 1, k >= 1, in which row n lists the partitions of n into consecutive parts, with the partitions ordered by increasing number of parts.
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31
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1, 2, 3, 2, 1, 4, 5, 3, 2, 6, 3, 2, 1, 7, 4, 3, 8, 9, 5, 4, 4, 3, 2, 10, 4, 3, 2, 1, 11, 6, 5, 12, 5, 4, 3, 13, 7, 6, 14, 5, 4, 3, 2, 15, 8, 7, 6, 5, 4, 5, 4, 3, 2, 1, 16, 17, 9, 8, 18, 7, 6, 5, 6, 5, 4, 3, 19, 10, 9, 20, 6, 5, 4, 3, 2, 21, 11, 10, 8, 7, 6, 6, 5, 4, 3, 2, 1, 22, 7, 6, 5, 4, 23, 12, 11
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OFFSET
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1,2
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COMMENTS
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In the triangle the first partition with m parts appears as the last partition in row A000217(m), m >= 1. - Omar E. Pol, Mar 23 2022
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
[1];
[2];
[3], [2, 1];
[4];
[5], [3, 2];
[6], [3, 2, 1];
[7], [4, 3];
[8];
[9], [5, 4], [4, 3, 2];
[10], [4, 3, 2, 1];
[11], [6, 5];
[12], [5, 4, 3];
[13], [7, 6];
[14], [5, 4, 3, 2];
[15], [8, 7], [6, 5, 4], [5, 4, 3, 2, 1];
[16];
[17], [9, 8];
[18], [7, 6, 5], [6, 5, 4, 3];
[19], [10, 9];
[20], [6, 5, 4, 3, 2];
[21], [11, 10], [8, 7, 6], [6, 5, 4, 3, 2, 1];
[22], [7, 6, 5, 4];
[23], [12, 11];
[24], [9, 8, 7];
[25], [13, 12], [7, 6, 5, 4, 3];
[26], [8, 7, 6, 5];
[27], [14, 13], [10, 9, 8], [7, 6, 5, 4, 3, 2];
[28], [7, 6, 5, 4, 3, 2, 1];
...
Note that in the below diagram the number of horizontal line segments in the n-th row equals A001227(n), the number of partitions of n into consecutive parts, so we can find the partitions of n into consecutive parts as follows: consider the vertical blocks of numbers that start exactly in the n-th row of the diagram, for example: for n = 15 consider the vertical blocks of numbers that start exactly in the 15th row. They are [15], [8, 7], [6, 5, 4] and [5, 4, 3, 2, 1], equaling the 15th row of the above triangle.
. _
. _|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 diagram is infinite. For more information about the diagram see A286000.
For an amazing connection with sum of divisors function (A000203) see A237593.
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MATHEMATICA
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intervals[n_]:=Module[{x, y}, SolveValues[(x^2-y^2+x+y)/2==n&&0<x<=n&&0<y<=n, {x, y}, Integers]];
A299765row[n_]:=Flatten[SortBy[Map[Range[First[#], Last[#], -1]&, intervals[n]], Length]];
nrows=25; Array[A299765row, nrows] (* Paolo Xausa, Jun 19 2022 *)
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PROG
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(PARI) iscons(p) = my(v = vector(#p-1, k, p[k+1] - p[k])); v == vector(#p-1, i, 1);
row(n) = my(list = List()); forpart(p=n, if (iscons(p), listput(list, Vecrev(p))); ); Vec(list); \\ Michel Marcus, May 11 2022
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CROSSREFS
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The number of partitions into consecutive parts in row n is A001227(n).
For tables of partitions into consecutive parts see A286000 and A286001.
Cf. A000203, A000217, A026792, A235791, A237048, A237591, A237593, A245092, A285914, A286013, A288529, A288772, A288773, A288774.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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