A228525 -id:A228525 - cp's OEIS frontend

A236000 Triangle read by rows in which row n lists the overpartitions of n in colexicographic order.

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 2, 2, 2, 2, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1

Offset: 1

Omar E. Pol, Jan 18 2014

In the data section the overlined parts cannot be represented correctly, therefore the sequence represents all possible suborderings generated by the overlined parts.
The diagram in the second part of the Example section shows only one of the possible suborderings.
The equivalent sequence for partitions is A211992.
The equivalent sequence for compositions is A228525.
See both sequences for more information.
Row n contains A015128(n) overpartitions.
Row n contains A235792(n) parts.
Row sums give A235793.

			Triangle begins:
[1], [1];
[1, 1], [1, 1], [2], [2];
[1, 1, 1], [1, 1, 1], [2, 1], [2, 1], [2, 1], [2, 1], [3], [3];
[1, 1, 1, 1], [1, 1, 1, 1], [2, 1, 1], [2, 1, 1], [2, 1, 1], [2, 1, 1], [3, 1], [3, 1], [3, 1], [3, 1], [2, 2], [2, 2], [4], [4];
...
Illustration of initial terms (n: 1..4)
-----------------------------------------
n      Diagram          Overpartition
-----------------------------------------
.       _
1      |.|              1',
1      |_|              1;
.       _ _
2      |.| |            1', 1,
2      |_| |            1,  1,
2      |  .|            2',
2      |_ _|            2;
.       _ _ _
3      |.| | |          1', 1,  1,
3      |_| | |          1,  1,  1,
3      |  .|.|          2', 1',
3      |   |.|          2,  1',
3      |  .| |          2', 1,
3      |_ _| |          2,  1,
3      |    .|          3',
3      |_ _ _|          3;
.       _ _ _ _
4      |.| | | |        1', 1,  1,  1,
4      |_| | | |        1,  1,  1,  1,
4      |  .|.| |        2', 1', 1,
4      |   |.| |        2,  1', 1,
4      |  .| | |        2', 1,  1,
4      |_ _| | |        2,  1,  1,
4      |    .|.|        3', 1',
4      |     |.|        3,  1',
4      |    .| |        3', 1,
4      |_ _ _| |        3,  1,
4      |  .|   |        2', 2,
4      |_ _|   |        2,  2,
4      |      .|        4',
4      |_ _ _ _|        4;
.

Cf. A015128, A211992, A228525, A235790, A235792, A235793, A235797, A235798, A236001

A228350 Triangle read by rows: T(j,k) is the k-th part in nonincreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 6, 5, 4, 4, 3, 3

Offset: 1

Omar E. Pol, Aug 20 2013

Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A065120, n >= 1.

The equivalent sequence for integer partitions is A206437.

			---------------------------------------------------------
.              Diagram                Triangle
Compositions     of            of compositions (rows)
.   of 5       regions          and regions (columns)
----------------------------------------------------------
.             _ _ _ _ _
.         5  |_        |                                5
.       1+4  |_|_      |                              1 4
.       2+3  |_  |     |                            2   3
.     1+1+3  |_|_|_    |                          1 1   3
.       3+2  |_    |   |                        3       2
.     1+2+2  |_|_  |   |                      1 2       2
.     2+1+2  |_  | |   |                    2   1       2
.   1+1+1+2  |_|_|_|_  |                  1 1   1       2
.       4+1  |_      | |                4               1
.     1+3+1  |_|_    | |              1 3               1
.     2+2+1  |_  |   | |            2   2               1
.   1+1+2+1  |_|_|_  | |          1 1   2               1
.     3+1+1  |_    | | |        3       1               1
.   1+2+1+1  |_|_  | | |      1 2       1               1
.   2+1+1+1  |_  | | | |    2   1       1               1
. 1+1+1+1+1  |_|_|_|_|_|  1 1   1       1               1
.
Also the structure could be represented by an isosceles triangle in which the n-th diagonal gives the n-th region. For the composition of 4 see below:
.             _ _ _ _
.         4  |_      |                  4
.       1+3  |_|_    |                1   3
.       2+2  |_  |   |              2       2
.     1+1+2  |_|_|_  |            1   1       2
.       3+1  |_    | |          3               1
.     1+2+1  |_|_  | |        1   2               1
.     2+1+1  |_  | | |      2       1               1
.   1+1+1+1  |_|_|_|_|    1   1       1               1
.
Illustration of the four sections of the set of compositions of 4:
.                                      _ _ _ _
.                                     |_      |     4
.                                     |_|_    |   1+3
.                                     |_  |   |   2+2
.                       _ _ _         |_|_|_  | 1+1+2
.                      |_    |   3          | |     1
.             _ _      |_|_  | 1+2          | |     1
.     _      |_  | 2       | |   1          | |     1
.    |_| 1     |_| 1       |_|   1          |_|     1
.
.
Illustration of initial terms. The parts of the eight regions of the set of compositions of 4:
--------------------------------------------------------
\j:  1      2    3        4     5      6    7          8
k
--------------------------------------------------------
.  _    _ _    _    _ _ _     _    _ _    _    _ _ _ _
1 |_|1 |_  |2 |_|1 |_    |3  |_|1 |_  |2 |_|1 |_      |4
2        |_|1        |_  |2         |_|1        |_    |3
3                      | |1                       |   |2
4                      |_|1                       |_  |2
5                                                   | |1
6                                                   | |1
7                                                   | |1
8                                                   |_|1
.
Triangle begins:
1;
2,1;
1;
3,2,1,1;
1;
2,1;
1;
4,3,2,2,1,1,1,1;
1;
2,1;
1;
3,2,1,1;
1;
2,1;
1;
5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1;
...
.
Also triangle read by rows T(n,m) in which row n lists the parts of the n-th section of the set of compositions of the integers >= n, ordered by regions. Row lengths give A045623. Row sums give A001792 (see below):
[1];
[2,1];
[1],[3,2,1,1];
[1],[2,1],[1],[4,3,2,2,1,1,1,1];
[1],[2,1],[1],[3,2,1,1],[1],[2,1],[1],[5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1];

Main triangle: column 1 is A001511. Row j has length A006519(j). Row sums give A038712.

Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A135010, A141285, A187816, A187818, A193870, A206437, A228347, A228348, A228349, A228351, A228366, A228367, A228370, A228371, A228525, A228526.

T(j,k) = A065120(A001511(j)),k) = A001511(j) - A029837(k), 1<=k<=A006519(j), j>=1.

A228526 Triangle read by rows: T(n,k) = sum of all parts of size k in all compositions (ordered partitions) of n.

Original entry on oeis.org

1, 2, 2, 5, 4, 3, 12, 10, 6, 4, 28, 24, 15, 8, 5, 64, 56, 36, 20, 10, 6, 144, 128, 84, 48, 25, 12, 7, 320, 288, 192, 112, 60, 30, 14, 8, 704, 640, 432, 256, 140, 72, 35, 16, 9, 1536, 1408, 960, 576, 320, 168, 84, 40, 18, 10, 3328, 3072, 2112, 1280, 720

Offset: 1

Omar E. Pol, Aug 28 2013

The equivalent sequence for partitions is A138785, see the first comment there.

			T(4,2) = 10 because there are 5 parts of size 2 in all compositions of 4, T(4,2) = 5*2 = 10 (see below):
---------------------------------------------------------
. Compositions                   Parts      Sum of parts
.     of 4        Diagram      of size 2     of size 2
---------------------------------------------------------
.                 _ _ _ _
.   1+1+1+1      |_| | | |         0             0
.     2+1+1      |_ _| | |         1             2
.     1+2+1      |_|   | |         1             2
.       3+1      |_ _ _| |         0             0
.     1+1+2      |_| |   |         1             2
.       2+2      |_ _|   |         2             4
.       1+3      |_|     |         0             0
.         4      |_ _ _ _|         0             0
.                                -----        ------
.                           Total  5            10
.
Triangle begins:
1;
2,       2;
5,       4,    3;
12,     10,    6,    4;
28,     24,   15,    8,   5;
64,     56,   36,   20,  10,   6;
144,   128,   84,   48,  25,  12,   7;
320,   288,  192,  112,  60,  30,  14,  8;
704,   640,  432,  256, 140,  72,  35, 16,  9;
1536, 1408,  960,  576, 320, 168,  84, 40, 18, 10;
3328, 3072, 2112, 1280, 720, 384, 196, 96, 45, 20, 11;
...

Column k is k*A045623. Row sums give A001787, n >= 1. Right border gives A000027.

Cf. A001792, A011782, A138785, A221876, A228525, A228527.

T(n,k) = k*A045623(n-k) = k*A221876(n,k), n >=1, 1<=k<=n.

A187816 Triangle read by rows in which row n lists the first 2^(n-1) terms of A006519 in nonincreasing order, n >= 1.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 16, 8, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 32, 16, 8, 8, 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 64, 32, 16, 16, 8, 8, 8, 8, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 2, 2

Offset: 1

Omar E. Pol, Sep 10 2013

T(n,k) is also the number of parts in the k-th largest region of the diagram of regions of the set of compositions of n, n >= 1, k >= 1, see example.
Row lengths is A000079.
Row sums give A001792(n-1).

			For n = 5 the diagram of regions of the set of compositions of 5 has 2^(5-1) regions, see below:
------------------------------------------------------
.          A006519
.         as a tree
.         of number        Diagram
Region    of parts       of regions     Composition
------------------------------------------------------
.                         _ _ _ _ _
1      | 1          |    |_| | | | |    1, 1, 1, 1, 1
2      |   2        |    |_ _| | | |    2, 1, 1, 1
3      | 1          |    |_|   | | |    1, 2, 1, 1
4      |      4     |    |_ _ _| | |    3, 1, 1
5      | 1          |    |_| |   | |    1, 1, 2, 1
6      |   2        |    |_ _|   | |    2, 2, 1
7      | 1          |    |_|     | |    1, 3, 1
8      |        8   |    |_ _ _ _| |    4, 1
9      | 1          |    |_| | |   |    1, 1, 1, 2
10     |   2        |    |_ _| |   |    2, 1, 2
11     | 1          |    |_|   |   |    1, 2, 2
12     |      4     |    |_ _ _|   |    3, 2
13     | 1          |    |_| |     |    1, 1, 3
14     |   2        |    |_ _|     |    2, 3
15     | 1          |    |_|       |    1, 4
16     |         16 |    |_ _ _ _ _|    5
.
The first largest region in the diagram is the 16th region which contains 16 parts, so T(5,1) = 16. The second largest region is the 8th region which contains 8 parts, so T(5,2) = 8. The third and the fourth largest regions are both the 4th region and the 12th region, each contains 4 parts, so T(5,3) = 4 and T(5,4) = 4. And so on. The sequence of the number of parts of the k-th largest region of the diagram is [16, 8, 4, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1], the same as the 5th row of triangle, as shown below.
Triangle begins:
1;
2,1;
4,2,1,1;
8,4,2,2,1,1,1,1;
16,8,4,4,2,2,2,2,1,1,1,1,1,1,1,1;
32,16,8,8,4,4,4,4,2,2,2,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
...

Cf. A000079, A001511, A001792, A006519, A011782, A065120, A187818, A228525, A228369.

A187818 Triangle read by rows in which row n lists the first 2^(n-1) terms of A038712 in nonincreasing order, n >= 1.

Original entry on oeis.org

1, 3, 1, 7, 3, 1, 1, 15, 7, 3, 3, 1, 1, 1, 1, 31, 15, 7, 7, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 63, 31, 15, 15, 7, 7, 7, 7, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 127, 63, 31, 31, 15, 15, 15, 15, 7, 7, 7, 7, 7, 7, 7, 7, 3, 3

Offset: 1

Omar E. Pol, Sep 10 2013

T(n,k) is also the sum of all parts of the k-th largest region of the diagram of regions of the set of compositions of n, n >= 1, k >= 1, see example.
Row lengths is A000079.
Row sums give A001787, n >= 1.

			For n = 5 the diagram of regions of the set of compositions of 5 has 2^(5-1) regions, see below:
------------------------------------------------------
.         A038712 as
.       a tree of sum      Diagram
Region   of all parts    of regions     Composition
------------------------------------------------------
.                         _ _ _ _ _
1      | 1          |    |_| | | | |    1, 1, 1, 1, 1
2      |   3        |    |_ _| | | |    2, 1, 1, 1
3      | 1          |    |_|   | | |    1, 2, 1, 1
4      |      7     |    |_ _ _| | |    3, 1, 1
5      | 1          |    |_| |   | |    1, 1, 2, 1
6      |   3        |    |_ _|   | |    2, 2, 1
7      | 1          |    |_|     | |    1, 3, 1
8      |       15   |    |_ _ _ _| |    4, 1
9      | 1          |    |_| | |   |    1, 1, 1, 2
10     |   3        |    |_ _| |   |    2, 1, 2
11     | 1          |    |_|   |   |    1, 2, 2
12     |      7     |    |_ _ _|   |    3, 2
13     | 1          |    |_| |     |    1, 1, 3
14     |   3        |    |_ _|     |    2, 3
15     | 1          |    |_|       |    1, 4
16     |         31 |    |_ _ _ _ _|    5
.
The first largest region in the diagram is the 16th region which contains 16 parts and the sum of parts is 31, so T(5,1) = 31. The second largest region is the 8th region which contains 8 parts and the sum of parts is 15, so T(5,2) = 15. The third and the fourth largest regions are both the 4th region and the 12th region, each contains 4 parts and the sum of parts is 7, so T(5,3) = 7 and T(5,4) = 7. And so on. The sequence of the sum of all parts of the k-th largest region of the diagram is [31, 15, 7, 7, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1], the same as the 5th row of triangle, as shown below.
Triangle begins:
1;
3,1;
7,3,1,1;
15,7,3,3,1,1,1,1;
31,15,7,7,3,3,3,3,1,1,1,1,1,1,1,1;
63,31,15,15,7,7,7,7,3,3,3,3,3,3,3,3,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1;
...

Cf. A000079, A000225, A001511, A001787, A001792, A006519, A011782, A038712, A065120, A187816, A228525, A228369.

A228349 Triangle read by rows: T(j,k) is the k-th part in nondecreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 3, 4, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 3, 4, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Aug 26 2013

Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A090996, n >= 1.

The equivalent sequence for partitions is A220482.

			----------------------------------------------------------
.             Diagram                Triangle
Compositions    of            of compositions (rows)
of 5          regions          and regions (columns)
----------------------------------------------------------
.            _ _ _ _ _
5           |_        |                                 5
1+4         |_|_      |                               1 4
2+3         |_  |     |                             2   3
1+1+3       |_|_|_    |                           1 1   3
3+2         |_    |   |                         3       2
1+2+2       |_|_  |   |                       1 2       2
2+1+2       |_  | |   |                     2   1       2
1+1+1+2     |_|_|_|_  |                   1 1   1       2
4+1         |_      | |                 4               1
1+3+1       |_|_    | |               1 3               1
2+2+1       |_  |   | |             2   2               1
1+1+2+1     |_|_|_  | |           1 1   2               1
3+1+1       |_    | | |         3       1               1
1+2+1+1     |_|_  | | |       1 2       1               1
2+1+1+1     |_  | | | |     2   1       1               1
1+1+1+1+1   |_|_|_|_|_|   1 1   1       1               1
.
Written as an irregular triangle in which row n lists the parts of the n-th region the sequence begins:
1;
1,2;
1;
1,1,2,3;
1;
1,2;
1;
1,1,1,1,2,2,3,4;
1;
1,2;
1;
1,1,2,3;
1;
1,2;
1;
1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
...
Alternative interpretation of this sequence:
Triangle read by rows in which row r lists the regions of the last section of the set of compositions of r:
[1];
[1,2];
[1],[1,1,2,3];
[1],[1,2],[1],[1,1,1,1,2,2,3,4];
[1],[1,2],[1],[1,1,2,3],[1],[1,2],[1],[1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5];

Michael De Vlieger, Table of n, a(n) for n = 1..13312 (rows 1 <= n <= 2^11 = 2048).

Main triangle: Right border gives A001511. Row j has length A006519(j). Row sums give A038712.

Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A090996, A186114, A187816, A187818, A206437, A220482, A228347, A228348, A228350, A228351, A228366, A228367, A228370, A228371, A228525, A228526.

Table[Map[Length@ TakeWhile[IntegerDigits[#, 2], # == 1 &] &, Range[2^(# - 1), 2^# - 1]] &@ IntegerExponent[2 n, 2], {n, 32}] // Flatten (* Michael De Vlieger, May 23 2017 *)

A273140 Number of parts in the corner of size n X n of the modular table of partitions described in Comments.

Original entry on oeis.org

1, 3, 6, 11, 17, 25, 34, 46, 59, 74, 90, 109, 129, 151, 174, 201, 229, 259, 290, 323, 358, 394, 434, 475, 518, 562, 609, 657, 707, 758, 814, 871, 930, 990, 1052, 1116, 1181, 1249, 1318, 1389, 1462, 1536, 1615, 1695, 1777, 1860, 1946, 2033, 2122, 2212, 2305, 2400, 2496, 2594, 2694, 2795

Offset: 1

Omar E. Pol, May 16 2016

nonn

Consider an infinite dissection of the fourth quadrant of the square grid in which, apart from the axes x and y, the k-th horizontal line segment has length A141285(n) and the n-th vertical line segment has length A194446(n). Both line segments shares the point (A141285(n),n). Note that in the infinite table there are no partitions because every row contains an infinite number of parts. On the other hand, taking only the first k sections from the table we have a representation of the partitions of k. For the definition of "region" see A206437. For the definition of "section" see A135010.

			For n = 4 the corner of size 4 X 4 of the modular table of partitions contains 11 parts as shown below, so a(4) = 11.
.
.   Row   _ _ _ _       Parts
.    1   |_| | | |        4
.    2   |_ _| | |        3
.    3   |_ _ _| |        2
.    4   |_ _|   |        2
.                       ----
.                  Total 11
.
For n = 20 the corner of size 20 X 20 of the modular table of partitions contains 323 parts as shown below, so a(20) = 323.
.
.   Row   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _       Parts
.    1   |_| | | | | | | | | | | | | | | | | | | |        20
.    2   |_ _| | | | | | | | | | | | | | | | | | |        19
.    3   |_ _ _| | | | | | | | | | | | | | | | | |        18
.    4   |_ _|   | | | | | | | | | | | | | | | | |        18
.    5   |_ _ _ _| | | | | | | | | | | | | | | | |        17
.    6   |_ _ _|   | | | | | | | | | | | | | | | |        17
.    7   |_ _ _ _ _| | | | | | | | | | | | | | | |        16
.    8   |_ _|   |   | | | | | | | | | | | | | | |        17
.    9   |_ _ _ _|   | | | | | | | | | | | | | | |        16
.   10   |_ _ _|     | | | | | | | | | | | | | | |        16
.   11   |_ _ _ _ _ _| | | | | | | | | | | | | | |        15
.   12   |_ _ _|   |   | | | | | | | | | | | | | |        16
.   13   |_ _ _ _ _|   | | | | | | | | | | | | | |        15
.   14   |_ _ _ _|     | | | | | | | | | | | | | |        15
.   15   |_ _ _ _ _ _ _| | | | | | | | | | | | | |        14
.   16   |_ _|   |   |   | | | | | | | | | | | | |        16
.   17   |_ _ _ _|   |   | | | | | | | | | | | | |        15
.   18   |_ _ _|     |   | | | | | | | | | | | | |        15
.   19   |_ _ _ _ _ _|   | | | | | | | | | | | | |        14
.   20   |_ _ _ _ _|     | | | | | | | | | | | | |        14
.                                                       -----
.                                                  Total 323
.

Cf. A000041, A006128, A135010, A141285, A194446, A206437, A211992, A220482, A228525, A278602.

A337243 Compositions, sorted by increasing sum, increasing length, and increasing colexicographical order.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 3, 1, 2, 2, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 5, 4, 1, 3, 2, 2, 3, 1, 4, 3, 1, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 2, 2, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1

Offset: 1

Lorenzo Sauras Altuzarra, Aug 21 2020

			The first 5 rows are:
(1),
(2), (1, 1),
(3), (2, 1), (1, 2), (1, 1, 1),
(4), (3, 1), (2, 2), (1, 3), (2, 1, 1), (1, 2, 1), (1, 1, 2), (1, 1, 1, 1),
(5), (4, 1), (3, 2), (2, 3), (1, 4), (3, 1, 1), (2, 2, 1), (1, 3, 1), (2, 1, 2), (1, 2, 2), (1, 1, 3), (2, 1, 1, 1), (1, 2, 1, 1), (1, 1, 2, 1), (1, 1, 1, 2), (1, 1, 1, 1, 1).

Cf. A124734 (increasing length, then lexicographic).
Cf. A296774 (increasing length, then reverse lexicographic).
Cf. A337259 (increasing length, then reverse colexicographic).
Cf. A296773 (decreasing length, then lexicographic).
Cf. A296772 (decreasing length, then reverse lexicographic).
Cf. A337260 (decreasing length, then colexicographic).
Cf. A108244 (decreasing length, then reverse colexicographic).
Cf. A228369 (lexicographic).
Cf. A066099 (reverse lexicographic).
Cf. A228525 (colexicographic).
Cf. A228351 (reverse colexicographic).

List := proc(n)
   local i, j, k, L:
   L := []:
   for i from 1 to n do
      for j from 1 to i do
         L := [op(L), op(combinat:-composition(i, j))]:
      od:
   od:
   for k from 1 to numelems(L) do L[k] := ListTools:-Reverse(L[k]): od:
   L:
end:

A337259 Compositions, sorted by increasing sum, increasing length and decreasing colexicographical order.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 4, 1, 3, 2, 2, 3, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 5, 1, 4, 2, 3, 3, 2, 4, 1, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 3, 1, 2, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Lorenzo Sauras Altuzarra, Aug 21 2020

			The first 5 rows are:
(1),
(2), (1, 1),
(3), (1, 2), (2, 1), (1, 1, 1),
(4), (1, 3), (2, 2), (3, 1), (1, 1, 2), (1, 2, 1), (2, 1, 1), (1, 1, 1, 1),
(5), (1, 4), (2, 3), (3, 2), (4, 1), (1, 1, 3), (1, 2, 2), (2, 1, 2), (1, 3, 1), (2, 2, 1), (3, 1, 1), (1, 1, 1, 2), (1, 1, 2, 1), (1, 2, 1, 1), (2, 1, 1, 1), (1, 1, 1, 1, 1).

Cf. A124734 (increasing length, then lexicographic).
Cf. A296774 (increasing length, then reverse lexicographic).
Cf. A337243 (increasing length, then colexicographic).
Cf. A296773 (decreasing length, then lexicographic).
Cf. A296772 (decreasing length, then reverse lexicographic).
Cf. A337260 (decreasing length, then colexicographic).
Cf. A108244 (decreasing length, then reverse colexicographic).
Cf. A228369 (lexicographic).
Cf. A066099 (reverse lexicographic).
Cf. A228525 (colexicographic).
Cf. A228351 (reverse colexicographic).

List := proc(n)
   local i, j, k, L:
   L := []:
   for i from 1 to n do
      for j from 1 to i do
         L := [op(L), op(ListTools:-Reverse([op(combinat:-composition(i, j))]))]:
      od:
   od:
   for k from 1 to numelems(L) do L[k] := ListTools:-Reverse(L[k]): od:
   L:
end:

A337260 Compositions, sorted by increasing sum, decreasing length and increasing colexicographical order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 2, 2, 1, 3, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 1, 2, 2, 1, 1, 3, 4, 1, 3, 2, 2, 3, 1, 4, 5

Offset: 1

Lorenzo Sauras Altuzarra, Aug 21 2020

			The first 5 rows are:
(1),
(1, 1), (2),
(1, 1, 1), (2, 1), (1, 2), (3),
(1, 1, 1, 1), (2, 1, 1), (1, 2, 1), (1, 1, 2), (3, 1), (2, 2), (1, 3), (4),
(1, 1, 1, 1, 1), (2, 1, 1, 1), (1, 2, 1, 1), (1, 1, 2, 1), (1, 1, 1, 2), (3, 1, 1), (2, 2, 1), (1, 3, 1), (2, 1, 2), (1, 2, 2), (1, 1, 3), (4, 1), (3, 2), (2, 3), (1, 4), (5).

Cf. A124734 (increasing length, then lexicographic).
Cf. A296774 (increasing length, then reverse lexicographic).
Cf. A337243 (increasing length, then colexicographic).
Cf. A337259 (increasing length, then reverse colexicographic).
Cf. A296773 (decreasing length, then lexicographic).
Cf. A296772 (decreasing length, then reverse lexicographic).
Cf. A108244 (decreasing length, then reverse colexicographic).
Cf. A228369 (lexicographic).
Cf. A066099 (reverse lexicographic).
Cf. A228525 (colexicographic).
Cf. A228351 (reverse colexicographic).

List := proc(n)
   local i, j, k, L:
   L := []:
   for i from 1 to n do
      for j from 1 to i do
         L := [op(L), op(combinat:-composition(i, i-j+1))]:
      od:
   od:
   for k from 1 to numelems(L) do L[k] := ListTools:-Reverse(L[k]): od:
   L:
end:

cp's OEIS Frontend

A236000 Triangle read by rows in which row n lists the overpartitions of n in colexicographic order.

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A228350 Triangle read by rows: T(j,k) is the k-th part in nonincreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

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A228526 Triangle read by rows: T(n,k) = sum of all parts of size k in all compositions (ordered partitions) of n.

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A187816 Triangle read by rows in which row n lists the first 2^(n-1) terms of A006519 in nonincreasing order, n >= 1.

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A187818 Triangle read by rows in which row n lists the first 2^(n-1) terms of A038712 in nonincreasing order, n >= 1.

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A228349 Triangle read by rows: T(j,k) is the k-th part in nondecreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

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Mathematica

A273140 Number of parts in the corner of size n X n of the modular table of partitions described in Comments.

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A337243 Compositions, sorted by increasing sum, increasing length, and increasing colexicographical order.

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Maple

A337259 Compositions, sorted by increasing sum, increasing length and decreasing colexicographical order.

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Maple

A337260 Compositions, sorted by increasing sum, decreasing length and increasing colexicographical order.

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Maple