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A328365 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.

%I A328365 #44 May 29 2025 06:16:16
%S A328365 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,
%T A328365 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,
%U A328365 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
%N A328365 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.
%C A328365 For m >= 0, row 2^m consists of just one element (2^m). - _Paolo Xausa_, May 24 2025
%H A328365 Paolo Xausa, <a href="/A328365/b328365.txt">Table of n, a(n) for n = 1..10350</a> (rows 1..500 of triangle, flattened)
%e A328365 Triangle begins:
%e A328365   [1];
%e A328365   [2];
%e A328365   [1, 2], [3];
%e A328365   [4];
%e A328365   [2, 3], [5];
%e A328365   [1, 2, 3], [6];
%e A328365   [3, 4], [7];
%e A328365   [8];
%e A328365   [2, 3, 4], [4, 5], [9];
%e A328365   [1, 2, 3, 4], [10];
%e A328365   [5, 6], [11];
%e A328365   [3, 4, 5], [12];
%e A328365   [6, 7], [13];
%e A328365   [2, 3, 4, 5], [14];
%e A328365   [1, 2, 3, 4, 5], [4, 5, 6], [7, 8], [15];
%e A328365   [16];
%e A328365   [8, 9], [17];
%e A328365   [3, 4, 5, 6], [5, 6, 7], [18];
%e A328365   [9, 10], [19];
%e A328365   [2, 3, 4, 5, 6], [20];
%e A328365   [1, 2, 3, 4, 5, 6], [6, 7, 8], [10, 11], [21];
%e A328365   [4, 5, 6, 7], [22];
%e A328365   [11, 12], [23];
%e A328365   [7, 8, 9], [24];
%e A328365   [3, 4, 5, 6, 7], [12, 13], [25];
%e A328365   [5, 6, 7, 8], [26];
%e A328365   [2, 3, 4, 5, 6, 7], [8, 9, 10], [13, 14], [27];
%e A328365   [1, 2, 3, 4, 5, 6, 7], [28];
%e A328365   ...
%e A328365 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].
%e A328365 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.
%e A328365 Row        _
%e A328365   1       |1|_
%e A328365   2       |_ 2|_
%e A328365   3       |1|  3|_
%e A328365   4       |2|_   4|_
%e A328365   5       |_ 2|    5|_
%e A328365   6       |1|3|_     6|_
%e A328365   7       |2|  3|      7|_
%e A328365   8       |3|_ 4|_       8|_
%e A328365   9       |_ 2|  4|        9|_
%e A328365   10      |1|3|  5|_        10|_
%e A328365   11      |2|4|_   5|         11|_
%e A328365   12      |3|  3|  6|_          12|_
%e A328365   13      |4|_ 4|    6|           13|_
%e A328365   14      |_ 2|5|_   7|_            14|_
%e A328365   15      |1|3|  4|    7|             15|_
%e A328365   16      |2|4|  5|    8|_              16|_
%e A328365   17      |3|5|_ 6|_     8|               17|_
%e A328365   18      |4|  3|  5|    9|_                18|_
%e A328365   19      |5|_ 4|  6|      9|                 19|_
%e A328365   20      |_ 2|5|  7|_    10|_                  20|_
%e A328365   21      |1|3|6|_   6|     10|                   21|_
%e A328365   22      |2|4|  4|  7|     11|_                    22|_
%e A328365   23      |3|5|  5|  8|_      11|                     23|_
%e A328365   24      |4|6|_ 6|    7|     12|_                      24|_
%e A328365   25      |5|  3|7|_   8|       12|                       25|_
%e A328365   26      |6|_ 4|  5|  9|_      13|_                        26|_
%e A328365   27      |_ 2|5|  6|    8|       13|                         27|_
%e A328365   28      |1|3|6|  7|    9|       14|                           28|
%e A328365   ...
%e A328365 The diagram is infinite. For more information about the diagram see A286001.
%e A328365 For an amazing connection with sum of divisors function (A000203) see A237593.
%t A328365 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 *)
%Y A328365 Mirror of A299765.
%Y A328365 Row n has length A204217(n).
%Y A328365 Row sums give A245579.
%Y A328365 Column 1 gives A118235.
%Y A328365 Right border gives A000027.
%Y A328365 Records give A000027.
%Y A328365 Where records occur gives A285899.
%Y A328365 The number of partitions into consecutive parts in row n is A001227(n).
%Y A328365 For tables of partitions into consecutive parts see A286000 and A286001.
%Y A328365 Cf. A000203, A026792, A235791, A237048, A237591, A237593, A245092, A285914, A286013, A288529, A288772, A288773, A288774, A328361, A328362.
%K A328365 nonn,tabf
%O A328365 1,2
%A A328365 _Omar E. Pol_, Oct 22 2019