A141285 -id:A141285 - cp's OEIS frontend

A182745 Second column of the table A182743.

2, 2, 3, 2, 3, 3, 4, 2, 3, 4, 3, 4, 5, 2, 3, 3, 4, 5, 3, 4, 4, 5, 6, 2, 3, 3, 4, 3, 4, 4, 5, 6, 3, 4, 5, 4, 5, 5, 6, 7, 2, 3, 4, 3, 4, 5, 3, 4, 5, 4, 5, 6, 7, 3, 4, 4, 5, 6, 4, 5, 6, 5, 6, 7, 8, 2, 3, 3, 4, 5, 3, 4, 4, 5, 6, 3, 4, 4, 5, 6, 4, 5, 5, 6, 7, 8, 3, 4, 5, 4, 5, 5, 6, 7, 4, 5, 6, 7, 5, 6, 6, 7, 8, 9, 2, 3, 3, 4, 3, 4, 4, 5, 6, 3, 4, 5, 4, 5, 5

Offset: 1

Omar E. Pol, Nov 30 2010

nonn

The second largest of the n-th partition of the table A182743.

Cf. A135010, A138121, A141285, A182731, A182733, A182742, A182743, A182744.

A182985 Largest part of the n-th row of the table version "mirror" of the shell model of partitions of A135010 and A182980.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 02 2011

If n is odd then row n lists the numbers of row n of A141285. If n is even then row n lists the numbers of row n of A141285 but in reverse order.

			Triangle begins:
1,
2,
3,
4, 2,
3, 5,
6, 3, 4, 2,
3, 5, 4, 7,
8, 4, 5, 6, 3, 4, 2,
3, 5, 4, 7, 3, 6, 5, 9,
10,5, 6, 7, 4, 8, 4, 5, 6, 3, 4, 2,
3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8, 7, 6, 11,

Cf. A135010, A181285, A182982, A182983.

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).

A194853 Pair(i,j): a folded version of the shell model of partitions (see comments for precise definition).

Original entry on oeis.org

0, 1, 0, 2, -2, 1, 0, 2, 0, 4, -1, 2, -4, 1, 0, 2, -2, 2, 0, 3, 0, 6

Offset: 1

Omar E. Pol, Feb 12 2012

In the shell model of partitions (Cf. A135010, A138121) the n-th largest part has length = A141285(n). After the folding of the partitions the pair(i,j) shows the horizontal location of the n-th largest part at the level n of the structure (see example).

			-------------------------------------------
                  Pair      The structure
n   A141285(n)   (i, j)    looks like this
-------------------------------------------
1       1         0, 1,           *
2       2         0, 2,           **
3       3        -2, 1,         ***
4       2         0, 2,           **
5       4         0, 4,           ****
6       3        -1, 2,          ***
7       5        -4, 1        *****
8       2         0, 2            **
9       4        -2, 2          ****
10      3         0, 3            ***
11      6         0, 6            ******

Omar E. Pol, Illustration of the shell model of partitions before of the folding of all partitions (for integers = 1..12)

Cf. A135010, A138121, A141285.

A141285(n) = abs(i(n)) + j(n).

A225597 Triangle read by rows: T(n,k) = total number of parts of all regions of the set of partitions of n whose largest part is k.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 3, 3, 5, 1, 3, 4, 5, 7, 1, 4, 5, 7, 7, 11, 1, 4, 6, 8, 9, 11, 15, 1, 5, 7, 11, 10, 15, 15, 22, 1, 5, 9, 12, 13, 17, 19, 22, 30, 1, 6, 10, 16, 15, 22, 21, 29, 30, 42, 1, 6, 12, 18, 19, 25, 26, 32, 38, 42, 56, 1, 7, 14, 23, 22, 33, 29, 41, 42, 54, 56, 77

Offset: 1

Omar E. Pol, Aug 02 2013

For the definition of "region" see A206437.

T(n,k) is also the number of parts that end in the k-th column of the diagram of regions of the set of partitions of n (see Example section).

			For n = 5 and k = 3 the set of partitions of 5 contains two regions whose largest part is 3, they are third region which contains three parts [3, 1, 1] and the sixth region which contains only one part [3]. Therefore the total number of parts is 3 + 1 = 4, so T(5,3) = 4.
.
.    Diagram    Illustration of parts ending in column k:
.    for n=5      k=1   k=2     k=3       k=4        k=5
.   _ _ _ _ _                                  _ _ _ _ _
.  |_ _ _    |                _ _ _           |_ _ _ _ _|
.  |_ _ _|_  |               |_ _ _|  _ _ _ _       |_ _|
.  |_ _    | |          _ _          |_ _ _ _|        |_|
.  |_ _|_  | |         |_ _|  _ _ _      |_ _|        |_|
.  |_ _  | | |          _ _  |_ _ _|       |_|        |_|
.  |_  | | | |      _  |_ _|     |_|       |_|        |_|
.  |_|_|_|_|_|     |_|   |_|     |_|       |_|        |_|
.
k = 1 2 3 4 5
.
The 5th row lists:  1     3       4         5          7
.
Triangle begins:
1;
1,  2;
1,  2,  3;
1,  3,  3,  5;
1,  3,  4,  5,  7;
1,  4,  5,  7,  7, 11;
1,  4,  6,  8,  9, 11, 15;
1,  5,  7, 11, 10, 15, 15, 22;
1,  5,  9, 12, 13, 17, 19, 22, 30;
1,  6, 10, 16, 15, 22, 21, 29, 30, 42;
1,  6, 12, 18, 19, 25, 26, 32, 38, 42, 56;
1,  7, 14, 23, 22, 33, 29, 41, 42, 54, 56, 77;

Column 1 is A000012. Column 2 are the numbers => 2 of A008619. Row sums give A006128, n>=1. Right border gives A000041, n>=1. Second right border gives A000041, n>=1.

Cf. A006128, A133041, A135010, A138137, A139582, A141285, A182377, A186114, A186412, A187219, A193870, A194446, A206437, A207779, A211978, A220517, A225598, A225600, A225610.

A299775 Irregular triangle read by rows in which row n lists the indices of the partitions into consecutive parts in the list of colexicographically ordered partitions of n.

Original entry on oeis.org

1, 2, 2, 3, 5, 6, 7, 6, 11, 14, 15, 22, 25, 29, 30, 25, 42, 55, 56

Offset: 1

Omar E. Pol, Mar 29 2018

If n > 1 and n is odd then row n ending in [p(n) - 1, p(n)], where p(n) is A000041(n).

			Triangle begins:
   1;
   2;
   2,  3;
   5;
   6,  7;
   6, 11;
  14, 15;
  22;
  25, 29, 30;
  25, 42;
  55, 56;
...
For n = 9 the partitions of 9 into consecutive parts are [4, 3, 2], [5, 4] and [9]. Then we have that in the list of colexicographically ordered partitions of 9 these partitions are in the rows 25, 29 and 30 respectively as shown below, so the 9th row of the triangle is [25, 29, 30].
--------------------------------------------------------
   p         Diagram          Partitions of 9
--------------------------------------------------------
        1 2 3 4 5 6 7 8 9
        _ _ _ _ _ _ _ _ _
   1   |_| | | | | | | | |   [1, 1, 1, 1, 1, 1, 1, 1, 1]
   2   |_ _| | | | | | | |   [2, 1, 1, 1, 1, 1, 1, 1]
   3   |_ _ _| | | | | | |   [3, 1, 1, 1, 1, 1, 1]
   4   |_ _|   | | | | | |   [2, 2, 1, 1, 1, 1, 1]
   5   |_ _ _ _| | | | | |   [4, 1, 1, 1, 1, 1]
   6   |_ _ _|   | | | | |   [3, 2, 1, 1, 1, 1]
   7   |_ _ _ _ _| | | | |   [5, 1, 1, 1, 1]
   8   |_ _|   |   | | | |   [2, 2, 2, 1, 1, 1]
   9   |_ _ _ _|   | | | |   [4, 2, 1, 1, 1]
  10   |_ _ _|     | | | |   [3, 3, 1, 1, 1]
  11   |_ _ _ _ _ _| | | |   [6, 1, 1, 1]
  12   |_ _ _|   |   | | |   [3, 2, 2, 1, 1]
  13   |_ _ _ _ _|   | | |   [5, 2, 1, 1]
  14   |_ _ _ _|     | | |   [4, 3, 1, 1]
  15   |_ _ _ _ _ _ _| | |   [7, 1, 1]
  16   |_ _|   |   |   | |   [2, 2, 2, 2, 1]
  17   |_ _ _ _|   |   | |   [4, 2, 2, 1]
  18   |_ _ _|     |   | |   [3, 3, 2, 1]
  19   |_ _ _ _ _ _|   | |   [6, 2, 1]
  20   |_ _ _ _ _|     | |   [5, 3, 1]
  21   |_ _ _ _|       | |   [4, 4, 1]
  22   |_ _ _ _ _ _ _ _| |   [8, 1]
  23   |_ _ _|   |   |   |   [3, 2, 2, 2]
  24   |_ _ _ _ _|   |   |   [5, 2, 2]
  25   |_ _ _ _|     |   |   [4, 3, 2]   <--- Consecutive parts
  26   |_ _ _ _ _ _ _|   |   [7, 2]
  27   |_ _ _|     |     |   [3, 3, 3]
  28   |_ _ _ _ _ _|     |   [6, 3]
  29   |_ _ _ _ _|       |   [5, 4]   <--- Consecutive parts
  30   |_ _ _ _ _ _ _ _ _|   [9]   <--- Consecutive parts
.

Row n has length A001227(n).
Right border gives A000041, n >= 1.
Cf. A211992 (partitions in colexicographic order).
Cf. A299765 (partitions into consecutive parts).
For tables of partitions into consecutive parts see also A286000 and A286001.
Cf. A000041, A135010, A141285, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A299474, A299475, A299773, A299774.

A182989 Partial sums of A182731.

Original entry on oeis.org

1, 4, 7, 12, 15, 20, 24, 31, 34, 39, 43, 50, 53, 59, 64, 75, 78, 83, 89

Offset: 1

Omar E. Pol, Jan 25 2011

nonn

I would like a table for this sequence. Then I would like to see the graphics!

Cf. A135010, A141285, A182731, A182727, A182988.

A185370 Triangle read by rows: T(n,k) is the number of occurrences of k in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 0, 1, 3, 1, 0, 1, 0, 0, 1, 5, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 7, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 11, 2, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 2, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 15, 4, 1, 1, 0, 0, 0, 1

Offset: 1

Omar E. Pol, Jan 25 2013

For the definition of "region of the set of partitions of j" see A206437.
T(n,k) is the number of occurrences of k in the n-th region of the shell model of partitions (see A135010).
T(n,k) is also the number of occurrences of k in the n-th row of triangles A186114, A193870, A206437 (and possibly more).
If the length of row n is a record then the length of row n is j and also A000041(j) = n.
If A000041(j) = n then the sum of the last A187219(j) elements of column k is A182703(j,k) and also the sum of all elements of column k is A066633(j,k).

			First seven regions of any integer >= 5 are
[1], [2,1], [3,1,1], [2], [4,2,1,1,1], [3], [5,2,1,1,1,1,1] (see illustrations, see also A206437). The 7th region contains five 1's, only one 2 and only one 5. There are no 3's. There are no 4's, so row 7 is [5, 1, 0, 0, 1].
-----------------------------------------
n    j  m    k : 1  2  3  4  5  6  7  8
-----------------------------------------
1    1  1        1;
2    2  1        1, 1;
3    3  1        2, 0, 1;
4    4  1        0, 1;
5    4  2        3, 1, 0, 1;
6    5  1        0, 0, 1;
7    5  2        5, 1, 0, 0, 1;
8    6  1        0, 1;
9    6  2        0, 1, 0, 1;
10   6  3        0, 0, 1;
11   6  4        7, 2, 1, 0, 0, 1;
12   7  1        0, 0, 1;
13   7  2        0, 1, 0, 0, 1;
14   7  3        0, 0, 0, 1;
15   7  4       11, 2, 1, 0, 0, 0, 1;
16   8  1        0, 1;
17   8  2        0, 1, 0, 1;
18   8  3        0, 0, 1;
19   8  4        0, 2, 1, 0, 0, 1;
20   8  5        0, 0, 0, 0, 1;
21   8  6        0, 0, 0, 1;
22   8  7       15, 4, 1, 1, 0, 0, 0, 1;

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

Row n has length A141285(n). Row sums give A194446. Positive terms of column 1 give A000041.

Cf. A006128, A066633, A135010, A182703, A186114, A187219, A193870, A206437.

A206441 Triangle read by rows. T(n,k) = number of distinct parts in the k-th region of the last section of the set of partitions of n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 13 2012

a(n) is also the number of distinct parts in the n-th region of the shell model of partitions (see A135010 and A206437).

			The first region in the last section of the set of partitions of 6 looks like this:
.        **
There is only one part, so T(6,1) = 1.
The second region in the last section of the set of partitions of 6 looks like this:
.        ****
.          **
There are two distinct parts, so T(6,2) = 2.
The third region in the last section of the set of partitions of 6 looks like this:
.        ***
There is only one part, so T(6,3) = 1.
The 4th region in the last section of the set of partitions of 6 looks like this:
.        ******
.           ***
.            **
.            **
.             *
.             *
.             *
.             *
.             *
.             *
.             *
There are four distinct parts, so T(6,4) = 4.
Written as a triangle:
1;
2;
2;
1, 3;
1, 3;
1, 2, 1, 4;
1, 2, 1, 4;
1, 2, 1, 3, 1, 1, 5;
1, 2, 1, 3, 1, 2, 1, 5;
1, 2, 1, 3, 1, 1, 4, 1, 2, 1, 1, 6;

Row n has length A187219(n).

Cf. A135010, A138121, A141285, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A194447.

A210451 n minus the number of parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j), with a(0) = 0.

Original entry on oeis.org

0, 0, 0, 0, 3, 0, 5, 0, 7, 7, 9, 0, 11, 11, 13, 0, 15, 15, 17, 15, 19, 20, 0, 22, 22, 24, 22, 26, 26, 28, 0, 30, 30, 32, 30, 34, 35, 30, 37, 37, 39, 40, 0, 42, 42, 44, 42, 46, 46, 48, 42, 50, 51, 50, 53, 54, 0, 56, 56, 58, 56, 60, 61, 56, 63, 63, 65, 66, 56, 68, 68, 70, 68, 72, 72, 74, 75, 0

Offset: 0

Omar E. Pol, Jan 27 2013

nonn

a(0) = 0 iff n is a partition number A000041, n >= 1.
a(n) is also the number of zeros in the n-th row of triangle A186114 and also of triangle A193870, n >= 1.
a(n) is also the value of the index "i" mentioned in the definition of "regions of the set of partitions" in A206437.

Cf. A135010, A141285, A194446, A186114, A193870, A194447, A206437.

a(n) = n - A194446(n) = n - A141285(n) + A194447(n).

a(n+1) = a(n+2) = n, if a(n) = 0 and n >= 7.

cp's OEIS Frontend

A182745 Second column of the table A182743.

Views

Author

Keywords

Comments

Crossrefs

A182985 Largest part of the n-th row of the table version "mirror" of the shell model of partitions of A135010 and A182980.

Views

Author

Keywords

Comments

Examples

Crossrefs

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A194853 Pair(i,j): a folded version of the shell model of partitions (see comments for precise definition).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A225597 Triangle read by rows: T(n,k) = total number of parts of all regions of the set of partitions of n whose largest part is k.

Views

Author

Keywords

Comments

Examples

Crossrefs

A299775 Irregular triangle read by rows in which row n lists the indices of the partitions into consecutive parts in the list of colexicographically ordered partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

A182989 Partial sums of A182731.

Views

Author

Keywords

Comments

Crossrefs

A185370 Triangle read by rows: T(n,k) is the number of occurrences of k in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A206441 Triangle read by rows. T(n,k) = number of distinct parts in the k-th region of the last section of the set of partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

A210451 n minus the number of parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j), with a(0) = 0.

Views

Author

Keywords

Comments

Crossrefs

Formula