A026792 -id:A026792 - cp's OEIS frontend

A211986 A list of certain compositions which arise from the ordered partitions of the positive integers in which the shells of each integer are arranged as the arms of a spiral.

1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 4, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 3, 5, 3, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 4, 1, 6, 3, 3, 2, 4, 2, 2, 2, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 2, 1, 5, 7, 4, 3, 5, 2, 3, 2, 2, 1, 5, 1, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 2, 2, 2, 1, 2, 4, 1, 3, 3, 1, 6, 1

Offset: 1

Omar E. Pol, Aug 19 2012

In order to construct this sequence we use the following rules:
- Consider the partitions of positive integers.
- For each positive integer its shells must be arranged as the arms of a spiral.
- The sequence lists one spiral for each positive integer.
- If the integer j is odd then the first composition listed of each spiral is j.
- If the integer j is even then we use the same spiral of A211988.

			----------------------------------------------
.                 Expanded         Geometric
Compositions     arrangement         model
----------------------------------------------
1;                    1;              |*|
----------------------------------------------
2;                  2 .;            |* *|
1,1;                1,1;            |*|o|
----------------------------------------------
3;                  . . 3;          |* * *|
1,1,1;              1,1,1;          |o|o|*|
2,1;                2 .,1;          |o o|*|
----------------------------------------------
4,;               4 . . .;        |* * * *|
2,2;              2 .,2 .;        |* *|* *|
1,2,1;            1,2 .,1;        |*|o o|o|
1,1,1,1,;         1,1,1,1;        |*|o|o|o|
1,3;              1,. . 3;        |*|o o o|
----------------------------------------------
5;                . . . . 5;      |* * * * *|
3,2;              . . 3,. 2;      |* * *|* *|
1,3,1;            1,. . 3,1;      |o|o o o|*|
1,1,1,1,1;        1,1,1,1,1;      |o|o|o|o|*|
1,2,1,1;          1,2 .,1,1;      |o|o o|o|*|
2,2,1;            2 .,2 .,1;      |o o|o o|*|
4,1;              4 . . .,1;      |o o o o|*|
----------------------------------------------
6;              6 . . . . .;    |* * * * * *|
3,3;            3 . .,3 . .;    |* * *|* * *|
2,4;            2 .,4 . . .;    |* *|* * * *|
2,2,2;          2 .,2 .,2 .;    |* *|* *|* *|
1,4,1;          1,4 . . .,1;    |*|o o o o|o|
1,2,2,1;        1,2 .,2 .,1;    |*|o o|o o|o|
1,1,2,1,1;      1,1,2 .,1,1;    |*|o|o o|o|o|
1,1,1,1,1,1;    1,1,1,1,1,1;    |*|o|o|o|o|o|
1,1,3,1;        1,1,. . 3,1;    |*|o|o o o|o|
1,3,2;          1,. . 3,. 2;    |*|o o o|o o|
1,5;            1,. . . . 5;    |*|o o o o o|
------------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Omar E. Pol, Illustration of the rows 30..44 (figure D)

Rows sums give A036042, n>=1.
Mirror of A211985. Other spiral versions are A211987, A211988, A211995-A211998. See also A026792, A211983, A211984, A211989, A211992, A211993, A211994, A211999.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A211987 A list of certain compositions which arise from the ordered partitions of the positive integers in which the shells of each integer are arranged as a spiral.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 18 2012

In order to construct this sequence we use the following rules:
- Consider the partitions of positive integers.
- For each positive integer its shells must be arranged in a spiral.
- The sequence lists one spiral for each positive integer.
- If the integer j is odd then the last composition listed of each spiral is j.
- If the integer j is even then the first composition listed of each spiral is j.
This sequence represents a three-dimensional structure in which each column contains parts of the same size.

			----------------------------------------------
.                Expanded        Geometric
Compositions    arrangement        model
----------------------------------------------
1;                  1;              |*|
----------------------------------------------
2;                  . 2;            |* *|
1,1;                1,1;            |o|*|
----------------------------------------------
1,2;              1,. 2;          |*|o o|
1,1,1;            1,1,1;          |*|o|o|
3;                3 . .;          |* * *|
----------------------------------------------
4,;               . . . 4;        |* * * *|
2,2;              . 2,. 2;        |* *|* *|
1,2,1;            1,. 2,1;        |o|o o|*|
1,1,1,1,;         1,1,1,1;        |o|o|o|*|
3,1;              3 . .,1;        |o o o|*|
----------------------------------------------
1,4;            1,. . . 4;      |*|o o o o|
1,2,2;          1,. 2,. 2;      |*|o o|o o|
1,1,2,1;        1,1,. 2,1;      |*|o|o o|o|
1,1,1,1,1;      1,1,1,1,1;      |*|o|o|o|o|
1,3,1;          1,3 . .,1;      |*|o o o|o|
2,3;            2 .,3 . .;      |* *|* * *|
5;              5 . . . .;      |* * * * *|
----------------------------------------------
6;              . . . . . 6;    |* * * * * *|
3,3;            . . 3,. . 3;    |* * *|* * *|
4,2;            . . . 4,. 2;    |* * * *|* *|
2,2,2;          . 2,. 2,. 2;    |* *|* *|* *|
1,4,1;          1,. . . 4,1;    |o|o o o o|*|
1,2,2,1;        1,. 2,. 2,1;    |o|o o|o o|*|
1,1,2,1,1;      1,1,. 2,1,1;    |o|o|o o|o|*|
1,1,1,1,1,1;    1,1,1,1,1,1;    |o|o|o|o|o|*|
1,3,1,1;        1,3 . .,1,1;    |o|o o o|o|*|
2,3,1;          2 .,3 . .,1;    |o o|o o o|*|
5,1;            5 . . . .,1;    |o o o o o|*|
----------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Omar E. Pol, Illustration of the transformation of the partitions of 7 (fig. A) into the rows 30..44 (fig. D)

Rows sums give A036042, n>=1.
Mirror of A211988. Other spiral versions are A211985, A211986, A211995-A211998. See also A026792, A211983, A211984, A211989, A211992, A211993, A211994, A211999.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 11 2021

			The sets are the columns below:
  1 2 1 3 1 2 1 4 1 2 3 1 1 2 1 5 1 2 3 4 1 1 1 2 2 3 1
      2   3 3 2   4 4 4 2 3 3 2   5 5 5 5 2 3 4 3 4 4 2
              3         4 4 4 3           5 5 5 5 5 5 3
                              4                       5
As a tetrangle, the first four triangles are:
  {1}
  {2},{1,2}
  {3},{1,3},{2,3},{1,2,3}
  {4},{1,4},{2,4},{3,4},{1,2,4},{1,3,4},{2,3,4},{1,2,3,4}

Wikiversity, Lexicographic and colexicographic order

Triangle lengths are A000079.
Triangle sums are A001793.
Positions of first appearances are A005183.
Set maxima are A070939.
Set lengths are A124736.
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A296774, A299755, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344085, A344086, A344087, A344088, A344089.
Partition/composition applications: A036043, A049085, A115623, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Mathematica
```
SortBy[Rest[Subsets[Range[5]]],Last]
```

A211028 A list of ordered partitions of the positive integers.

Original entry on oeis.org

1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 2, 2, 4, 3, 1, 2, 1, 1, 1, 1, 1, 1, 3, 2, 5, 2, 2, 1, 4, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 4, 2, 3, 3, 6, 3, 2, 1, 5, 1, 2, 2, 1, 1, 4, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 2, 5, 2, 4, 3, 7, 2, 2, 2, 1

Offset: 1

Omar E. Pol, Oct 30 2012

We start with 1. Then for each positive integer j the sequence lists the partitions of j that do not contain 1 as a part, in lexicographical order. Then the sequence lists the partitions of j-1 together with one part of size 1 as the last part of that partition. Then the partitions of j-2 together with two parts of size 1, and so on.

			Written as an irregular triangle, in which row n lists the partitions of n, the sequence begins:
  {1};
  {2},{1,1};
  {3},{2,1},{1,1,1};
  {2,2},{4},{3,1},{2,1,1},{1,1,1,1};
  {3,2},{5},{2,2,1},{4,1},{3,1,1},{2,1,1,1},{1,1,1,1,1};
  {2,2,2},{4,2},{3,3},{6},{3,2,1},{5,1},{2,2,1,1},{4,1,1},{3,1,1,1},{2,1,1,1,1},{1,1,1,1,1,1};
  ...

Cf. A026792, A135010.

cp's OEIS Frontend

A211986 A list of certain compositions which arise from the ordered partitions of the positive integers in which the shells of each integer are arranged as the arms of a spiral.

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A211987 A list of certain compositions which arise from the ordered partitions of the positive integers in which the shells of each integer are arranged as a spiral.

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Author

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Comments

Examples

Links

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A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

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Mathematica

A211028 A list of ordered partitions of the positive integers.

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Author

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Comments

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