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A328365
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Irregular triangle read by rows, T(n,k), n >= 1, k >= 1, in which row n lists in reverse order the partitions of n into consecutive parts.
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12
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1, 2, 1, 2, 3, 4, 2, 3, 5, 1, 2, 3, 6, 3, 4, 7, 8, 2, 3, 4, 4, 5, 9, 1, 2, 3, 4, 10, 5, 6, 11, 3, 4, 5, 12, 6, 7, 13, 2, 3, 4, 5, 14, 1, 2, 3, 4, 5, 4, 5, 6, 7, 8, 15, 16, 8, 9, 17, 3, 4, 5, 6, 5, 6, 7, 18, 9, 10, 19, 2, 3, 4, 5, 6, 20, 1, 2, 3, 4, 5, 6, 6, 7, 8, 10, 11, 21, 4, 5, 6, 7, 22, 11, 12, 23, 7, 8, 9, 24
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OFFSET
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1,2
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
[1];
[2];
[1, 2], [3];
[4];
[2, 3], [5];
[1, 2, 3], [6];
[3, 4], [7];
[8];
[2, 3, 4], [4, 5], [9];
[1, 2, 3, 4], [10];
[5, 6], [11];
[3, 4, 5], [12];
[6, 7], [13];
[2, 3, 4, 5], [14];
[1, 2, 3, 4, 5], [4, 5, 6], [7, 8], [15];
[16];
[8, 9], [17];
[3, 4, 5, 6], [5, 6, 7], [18];
[9, 10], [19];
[2, 3, 4, 5, 6], [20];
[1, 2, 3, 4, 5, 6], [6, 7, 8], [10, 11], [21];
[4, 5, 6, 7], [22];
[11, 12], [23];
[7, 8, 9], [24];
[3, 4, 5, 6, 7], [12, 13], [25];
[5, 6, 7, 8], [26];
[2, 3, 4, 5, 6, 7], [8, 9, 10], [13, 14], [27];
[1, 2, 3, 4, 5, 6, 7], [28];
...
For n = 9 there are three partitions of 9 into consecutive parts, they are [9], [5, 4], [4, 3, 2], so the 9th row of triangle is [2, 3, 4], [4, 5], [9].
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 [1, 2, 3, 4, 5], [4, 5, 6], [7, 8], [15], equaling the 15th row of the above triangle.
Row _
1 |1|_
2 |_ 2|_
3 |1| 3|_
4 |2|_ 4|_
5 |_ 2| 5|_
6 |1|3|_ 6|_
7 |2| 3| 7|_
8 |3|_ 4|_ 8|_
9 |_ 2| 4| 9|_
10 |1|3| 5|_ 10|_
11 |2|4|_ 5| 11|_
12 |3| 3| 6|_ 12|_
13 |4|_ 4| 6| 13|_
14 |_ 2|5|_ 7|_ 14|_
15 |1|3| 4| 7| 15|_
16 |2|4| 5| 8|_ 16|_
17 |3|5|_ 6|_ 8| 17|_
18 |4| 3| 5| 9|_ 18|_
19 |5|_ 4| 6| 9| 19|_
20 |_ 2|5| 7|_ 10|_ 20|_
21 |1|3|6|_ 6| 10| 21|_
22 |2|4| 4| 7| 11|_ 22|_
23 |3|5| 5| 8|_ 11| 23|_
24 |4|6|_ 6| 7| 12|_ 24|_
25 |5| 3|7|_ 8| 12| 25|_
26 |6|_ 4| 5| 9|_ 13|_ 26|_
27 |_ 2|5| 6| 8| 13| 27|_
28 |1|3|6| 7| 9| 14| 28|
...
The diagram is infinite. For more information about the diagram see A286001.
For an amazing connection with sum of divisors function (A000203) see A237593.
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MATHEMATICA
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Table[With[{h = Floor[n/2] - Boole[EvenQ@ n]}, Append[Array[Which[Total@ # == n, #, Total@ Most@ # == n, Most[#], True, Nothing] &@ NestWhile[Append[#, #[[-1]] + 1] &, {#}, Total@ # <= n &, 1, h - # + 1] &, h], {n}]], {n, 24}] // Flatten (* Michael De Vlieger, Oct 22 2019 *)
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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, A026792, A235791, A237048, A237591, A237593, A245092, A285914, A286013, A288529, A288772, A288773, A288774, A328361, A328362.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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