A138879 -id:A138879 - 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.

A228716 Triangle read by rows in which row n lists the rows (including 0's) of the n-th section of the set of partitions (in colexicographic order) of any integer >= n.

Original entry on oeis.org

1, 0, 1, 2, 0, 0, 1, 0, 1, 3, 0, 0, 0, 1, 0, 0, 1, 0, 1, 2, 2, 4, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 3, 2, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 2, 2, 2, 4, 2, 3, 3, 6, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Omar E. Pol, Sep 02 2013

In other words, row n lists the rows of the last section of the set of partitions (in colexicographic order) of n.
Row lengths is A006128.
The number of zeros in row n is A006128(n-1).
Rows sums give A138879.
For more properties of the sections of the set of partitions of a positive integer see example.
Positive terms give A230440. - Omar E. Pol, Oct 25 2013

			Illustration of the 15 rows of the 7th section (including zeros) of the set of partitions of any integer >= 7 (hence this is also the last section of the set of partitions of 7). Note that the sum of the k-th column is equal to the number of parts >= k, therefore the first differences of the column sums give the number of occurrences of parts k in the section. The same for all sections of all positive integers, see below:
-----------------------------
Column: 1  2  3  4  5  6  7
-----------------------------
Row |
1   |   0, 0, 0, 0, 0, 0, 1;
2   |   0, 0, 0, 0, 0, 1;
3   |   0, 0, 0, 0, 1;
4   |   0, 0, 0, 0, 1;
5   |   0, 0, 0, 1;
6   |   0, 0, 0, 1;
7   |   0, 0, 1;
8   |   0, 0, 0, 1;
9   |   0, 0, 1;
10  |   0, 0, 1;
11  |   0, 1;
12  |   3, 2, 2;
13  |   5, 2;
14  |   4, 3;
15  |   7;
-----------------------------
Sums:  19, 8, 5, 3, 2, 1, 1 -> Row 7 of triangle A207031.
.       | /| /| /| /| /| /|
.       |/ |/ |/ |/ |/ |/ |
F.Dif: 11, 3, 2, 1, 1, 0, 1 -> Row 7 of triangle A182703.
.
Triangle begins:
[1];
[0,1],[2];
[0,0,1],[0,1],[3];
[0,0,0,1],[0,0,1],[0,1],[2,2],[4];
[0,0,0,0,1],[0,0,0,1],[0,0,1],[0,0,1],[0,1],[3,2],[5];
[0,0,0,0,0,1],[0,0,0,0,1],[0,0,0,1],[0,0,0,1],[0,0,1],[0,0,1],[0,1],[2,2,2],[4,2],[3,3],[6];
[0,0,0,0,0,0,1],[0,0,0,0,0,1],[0,0,0,0,1],[0,0,0,0,1],[0,0,0,1],[0,0,0,1],[0,0,1],[0,0,0,1],[0,0,1],[0,0,1],[0,1],[3,2,2],[5,2],[4,3],[7];

Cf. A000041, A006128, A135010, A138121, A138879, A182703, A187219, A207031, A211992.

A350357 Irregular triangle read by rows in which row n lists all elements of the arrangement of the correspondence divisor/part related to the last section of the set of partitions of n in the following order: row n lists the n-th row of A138121 followed by the n-th row of A336812.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 26 2021

			Triangle begins:
[1], [1];
[2, 1], [1, 2];
[3, 1, 1], [1, 3, 1];
[4, 2, 2, 1, 1, 1], [1, 2, 4, 1, 2, 1];
[5, 3, 2, 1, 1, 1, 1, 1], [1, 5, 1, 3, 1, 2, 1, 1];
...
Illustration of the first six rows of triangle in an infinite table:
|---|---------|-----|-------|---------|-----------|-------------|---------------|
| n |         |  1  |   2   |    3    |     4     |      5      |       6       |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
|   |         |     |       |         |           |             |  6            |
|   |         |     |       |         |           |             |  3 3          |
|   |         |     |       |         |           |             |  4 2          |
| P |         |     |       |         |           |             |  2 2 2        |
| A |         |     |       |         |           |  5          |    1          |
| R |         |     |       |         |           |  3 2        |      1        |
| T |         |     |       |         |  4        |    1        |      1        |
| S |         |     |       |         |  2 2      |      1      |        1      |
|   |         |     |       |  3      |    1      |      1      |        1      |
|   |         |     |  2    |    1    |      1    |        1    |          1    |
|   |         |  1  |    1  |      1  |        1  |          1  |            1  |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
| D | A027750 |  1  |  1 2  |  1   3  |  1 2   4  |  1       5  |  1 2 3     6  |
| I | A027750 |     |       |  1      |  1 2      |  1   3      |  1 2   4      |
| V | A027750 |     |       |         |  1        |  1 2        |  1   3        |
| I | A027750 |     |       |         |           |  1          |  1 2          |
| S | A027750 |     |       |         |           |  1          |  1 2          |
| O | A027750 |     |       |         |           |             |  1            |
| R | A027750 |     |       |         |           |             |  1            |
| S |         |     |       |         |           |             |               |
|---|---------|-----|-------|---------|-----------|-------------|---------------|
.
For n = 6 in the upper zone of the above table we can see the parts of the last section of the set of partitions of 6 in reverse-colexicographic order in accordance with the 6th row of A138121.
In the lower zone of the table we can see the terms from the 6th row of A336812, these are the divisors of the numbers from the 6th row of A336811.
Note that in the lower zone of the table every row gives A027750.
The remarkable fact is that the elements in the lower zone of the arrangement are the same as the elements in the upper zone but in other order.
For an explanation of the connection of the elements of the upper zone with the elements of the lower zone, that is the correspondence divisor/part, see A336812 and A338156.
The growth of the upper zone of the table is in accordance with the growth of the modular prism described in A221529.
The growth of the lower zone of the table is in accordance with the growth of the tower described also in A221529.
The number of cubic cells added at n-th stage in each polycube is equal to A138879(10) = 150, hence the total number of cubic cells added at n-th stage is equal to 2*A138879(10) = 300, equaling the sum of the 10th row of this triangle.

Companion of A350333.
Row sums give 2*A138879.
Row lengths give 2*A138137.
Cf. A002865, A135010, A138121, A138879, A221529, A237593, A336812, A339278.

A139094 Largest part of the n-th row in the integrated diagram of the shell model of partitions.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 2, 3, 4, 6, 3, 4, 5, 7, 2, 3, 4, 4, 5, 6, 8, 3, 3, 4, 5, 5, 6, 7, 9, 2, 3, 4, 4, 4, 5, 5, 6, 6, 7, 8, 10, 3, 3, 4, 4, 5, 5, 5, 6, 6, 7, 7, 8, 9, 11, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 9, 10, 12

Offset: 1

Omar E. Pol, May 26 2008

nonn

Omar E. Pol, A shell model of partitions
Omar E. Pol, Illustration of initial terms

Cf. A138135, A138136, A138137, A138138, A138151, A138879, A138880, A139100.

A182285 Triangle read by rows: T(n,k) = sum of all parts in the k-th zone of the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 4, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 1, 1, 1, 1, 6, 6, 6, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 7, 7, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 8, 8, 8, 8, 8, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Apr 23 2012

Row n lists A000041(n-1) 1's together with A002865(n) n's.

			Illustration of three arrangements of the last section of the set of partitions of 7 and the zone numbers:
--------------------------------------------------------
Zone \   a)                    b)                    c)
--------------------------------------------------------
15      (7)                   (7)       (. . . . . . 7)
14      (4+3)               (4+3)       (. . . 4 . . 3)
13      (5+2)               (5+2)       (. . . . 5 . 2)
12      (3+2+2)           (3+2+2)       (. . 3 . 2 . 2)
11        (1)                 (1)                   (1)
10          (1)               (1)                   (1)
9           (1)               (1)                   (1)
8             (1)             (1)                   (1)
7           (1)               (1)                   (1)
6             (1)             (1)                   (1)
5             (1)             (1)                   (1)
4               (1)           (1)                   (1)
3               (1)           (1)                   (1)
2                 (1)         (1)                   (1)
1                   (1)       (1)                   (1)
.
For n = 7 and k = 12 we can see that in the 12th zone of the last section of 7 the parts are 3, 2, 2, therefore T(7,12) = 3+2+2 = 7.
Written as a triangle begins:
1;
1,2;
1,1,3;
1,1,1,4,4;
1,1,1,1,1,5,5;
1,1,1,1,1,1,1,6,6,6,6;
1,1,1,1,1,1,1,1,1,1,1,7,7,7,7;
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,8,8,8,8,8,8,8;

Row n has length A000041(n). Row sums give A138879.

Cf. A000041, A002865, A135010, A138121, A182284, A193173.

A194449 Largest part minus the number of parts > 1 in the n-th region of the set of partitions of j, if 1 <= n <= A000041(j).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 2, 2, 3, 3, 3, 1, 2, 2, 2, 4, 3, 1, 2, 3, 3, 3, 2, 4, 4, 1, 1, 2, 2, 2, 4, 3, 1, 3, 5, 5, 4, -2, 2, 3, 3, 3, 2, 4, 4, 1, 4, 3, 5, 6, 5, -3, 1, 2, 2, 2, 4, 3, 1, 3, 5, 5, 4, -2, 2, 4, 4, 5, 3, 6, 6, 5, -9

Offset: 1

Omar E. Pol, Dec 10 2011

Also triangle read by rows: T(j,k) = largest part minus the numbers of parts > 1 in the k-th region of the last section of the set of partitions of j. It appears that the sum of row j is equal to A000041(j-1). For the definition of "region" of the set of partitions of j see A206437. See also A135010.

			The 7th region of the shell model of partitions is [5, 2, 1, 1, 1, 1, 1]. The largest part is 5 and the number of parts > 1 is 2, so a(7) = 5 - 2 = 3 (see an illustration in the link section).
Written as an irregular triangle T(j,k) begins:
1;
1;
2;
1,2;
2,3;
1,2,2,2;
2,3,3,3;
1,2,2,2,4,3,1;
2,3,3,3,2,4,4,1;
1,2,2,2,4,3,1,3,5,5,4,-2;
2,3,3,3,2,4,4,1,4,3,5,6,5,-3;
1,2,2,2,4,3,1,3,5,5,4,-2,2,4,4,5,3,6,6,5,-9;

Omar E. Pol, Illustration of the seven regions of 5

Row j has length A187219(j).

Cf. A000041, A135010, A138121, A138137, A138879, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A194447, A206437.

a(n) = A141285(n) - A194448(n).

A194450 Vertex number of a rectangular spiral which contains exactly between its edges the successive shells of the partitions of the positive integers.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 17, 21, 28, 33, 44, 50, 65, 72, 94, 102, 132, 141, 183, 193, 249, 260, 337, 349, 450, 463, 598, 612, 788, 803, 1034, 1050, 1347, 1364, 1749, 1767, 2257, 2276, 2903, 2923, 3715, 3736, 4738, 4760, 6015, 6038, 7613, 7637, 9595

Offset: 0

Omar E. Pol, Nov 01 2011

nonn

First differences give A194451, the length of the edges of the spiral. For more information see A135010 and A138121.

Cf. A000041, A000217, A006128, A026905, A066186, A135010, A138121, A138879, A138880, A182703, A182994, A182995, A194451.

a(2n-1) = A026905(n) + A000217(n) - n, if n >= 1.

a(2n) = A026905(n) + A000217(n), if n >= 1.

A210946 Triangle read by rows: T(n,k) = sum of parts in the k-th column of the mirror of the last section of the set of partitions of n with its parts aligned to the right margin.

Original entry on oeis.org

1, 3, 5, 9, 2, 12, 3, 20, 9, 2, 25, 11, 3, 38, 22, 9, 2, 49, 28, 14, 3, 69, 44, 26, 9, 2, 87, 55, 37, 14, 3, 123, 83, 62, 29, 9, 2, 152

Offset: 1

Omar E. Pol, Apr 21 2012

Row n lists the positive terms of the n-th row of triangle A210953 in decreasing order.

			For n = 7 the illustration shows two arrangements of the last section of the set of partitions of 7:
.
.       (7)        (7)
.     (4+3)        (3+4)
.     (5+2)        (2+5)
.   (3+2+2)        (2+2+3)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.       (1)        (1)
.                 ---------
.                  25,11,3
.
The left hand picture shows the last section of 7 with its parts aligned to the right margin. In the right hand picture (the mirror) we can see that the sum of all parts of the columns 1..3 are 25, 11, 3 therefore row 7 lists 25, 11, 3.
Written as a triangle begins:
1;
3;
5;
9,    2;
12,   3;
20,   9,  2;
25,  11,  3;
38,  22,  9,  2;
49,  28, 14,  3;
69,  44, 26,  9,  2;
87,  55, 37, 14,  3,
123, 83, 62, 29,  9,  2;

Column 1 is A046746, n >= 1. Row sums give A138879.

Cf. A135010, A138121, A182703, A194714, A196807, A206437, A207031, A207034, A207035, A210945, A210952, A210953.

A220487 Partial sums of triangle A206437.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 11, 15, 17, 18, 19, 20, 23, 28, 30, 31, 32, 33, 34, 35, 37, 41, 43, 46, 52, 55, 57, 59, 60, 61, 62, 63, 64, 65, 66, 69, 74, 76, 80, 87, 90, 92, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 107, 111, 113, 116, 122, 125, 127, 129, 134, 138

Offset: 1

Omar E. Pol, Jan 18 2013

			When written as an irregular triangle in which row j has length A194446(j) then the right border gives A182244. Also the records of row lengths give the partition numbers (A000041) of the positive integers as shown below:
1;
3, 4;
7, 8, 9;
11;
15,17,18,19,20;
23;
28,30,31,32,33,34,35;
37;
41,43;
46;
52,55,57,59,60,61,62,63,64,65,66;
69;
74,76;
80;
87,90,92,94,95,96,97,98,99,100,101,102,103,104,105;
...
Also when written as an irregular triangle in which row j has length A138137(j) then the right border gives A066186 as shown below:
1;
3, 4;
7, 8, 9;
11,15,17,18,19,20;
23,28,30,31,32,33,34,35;
37,41,43,46,52,55,57,59,60,61,62,63,64,65,66;
69,74,76,80,87,90,92,94,95,96,97,98,99,100,101,102,103,104,105;
...

Cf. A000041, A006128, A066186, A135010, A138137, A138879, A141285, A182181, A182244, A186412, A194446, A206437.

a(A182181(n)) = A182244(n), n >= 1.

a(A006128(n)) = A066186(n), n >= 1.

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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A228716 Triangle read by rows in which row n lists the rows (including 0's) of the n-th section of the set of partitions (in colexicographic order) of any integer >= n.

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A350357 Irregular triangle read by rows in which row n lists all elements of the arrangement of the correspondence divisor/part related to the last section of the set of partitions of n in the following order: row n lists the n-th row of A138121 followed by the n-th row of A336812.

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A139094 Largest part of the n-th row in the integrated diagram of the shell model of partitions.

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A182285 Triangle read by rows: T(n,k) = sum of all parts in the k-th zone of the last section of the set of partitions of n.

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A194449 Largest part minus the number of parts > 1 in the n-th region of the set of partitions of j, if 1 <= n <= A000041(j).

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A194450 Vertex number of a rectangular spiral which contains exactly between its edges the successive shells of the partitions of the positive integers.

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A210946 Triangle read by rows: T(n,k) = sum of parts in the k-th column of the mirror of the last section of the set of partitions of n with its parts aligned to the right margin.

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A220487 Partial sums of triangle A206437.

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