A193870 -id:A193870 - cp's OEIS frontend

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).

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).

A194799 Triangle read by rows: T(n,k) = number of partitions of n that are formed by k sections, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 4, 2, 2, 1, 1, 1, 4, 4, 2, 2, 1, 1, 1, 7, 4, 4, 2, 2, 1, 1, 1, 8, 7, 4, 4, 2, 2, 1, 1, 1, 12, 8, 7, 4, 4, 2, 2, 1, 1, 1, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 1, 21, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 1, 24, 21, 14

Offset: 1

Omar E. Pol, Jan 30 2012

Also triangle read by rows in which row n lists the first k terms of A187219 in reverse order. For more information see A135010.

			Triangle begins:
1,
1,1,
1,1,1,
2,1,1,1,
2,2,1,1,1,
4,2,2,1,1,1,
4,4,2,2,1,1,1,
7,4,4,2,2,1,1,1,
8,7,4,4,2,2,1,1,1,
12,8,7,4,4,2,2,1,1,1,
14,12,8,7,4,4,2,2,1,1,1,
21,14,12,8,7,4,4,2,2,1,1,1

Columns are A187219. Row sums give A000041, n >= 1.

Cf. A002865, A116598, A135010, A138121, A182748, A186114, A193870, A194436-A194438, A194446.

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.

A244968 Area between two valleys at height 0 under the infinite Dyck path related to partitions in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, k >= 1, multiplied by 2.

Original entry on oeis.org

1, 4, 9, 28, 54, 151

Offset: 1

Omar E. Pol, Nov 08 2014

			For k = 6, the diagram 1 represents the partitions of 6. The diagram 2 is a minimalist version of the structure which does not contain the axes [X, Y], see below:
.
.  j     Diagram 1        Partitions          Diagram 2
.      _ _ _ _ _ _                           _ _ _ _ _ _
. 11  |_ _ _      |       6                  _ _ _      |
. 10  |_ _ _|_    |       3+3                _ _ _|_    |
.  9  |_ _    |   |       4+2                _ _    |   |
.  8  |_ _|_ _|_  |       2+2+2              _ _|_ _|_  |
.  7  |_ _ _    | |       5+1                _ _ _    | |
.  6  |_ _ _|_  | |       3+2+1              _ _ _|_  | |
.  5  |_ _    | | |       4+1+1              _ _    | | |
.  4  |_ _|_  | | |       2+2+1+1            _ _|_  | | |
.  3  |_ _  | | | |       3+1+1+1            _ _  | | | |
.  2  |_  | | | | |       2+1+1+1+1          _  | | | | |
.  1  |_|_|_|_|_|_|       1+1+1+1+1+1         | | | | | |
.
Then we use the elements from the above diagram to draw an infinite Dyck path in which the j-th odd-indexed segment has A141285(j) up-steps and the j-th even-indexed segment has A194446(j) down-steps.
For the illustration of initial terms we use two opposite Dyck paths, as shown below:
11 ...........................................................
.                                                            /\
.                                                           /
.                                                          /
7 ..................................                      /
.                                  /\                    /
5 ....................            /  \                /\/
.                    /\          /    \          /\  /
3 ..........        /  \        /      \        /  \/
2 .....    /\      /    \    /\/        \      /
1 ..  /\  /  \  /\/      \  /            \  /\/
0  /\/  \/    \/          \/              \/
.  \/\  /\    /\          /\              /\
.     \/  \  /  \/\      /  \            /  \/\
.   1      \/      \    /    \/\        /      \
.      4            \  /        \      /        \  /\
.           9        \/          \    /          \/  \
.                                 \  /                \/\
.                    28            \/                    \
.                                                         \
.                                  54                      \
.                                                           \
.                                                            \/
.
The diagram is infinite. Note that the n-th largest peak between two valleys at height 0 is also the partition number A000041(n).
Calculations:
a(1) = 1.
a(2) = 2^2 = 4.
a(3) = 3^2 = 9.
a(4) = 2^2-1^2+5^2 = 4-1+25 = 28.
a(5) = 3^2-2^2+7^2 = 9-4+49 = 54.
a(6) = 2^2-1^2+5^2-3^2+6^2-5^2+11^2 = 4-1+25-9+36-25+121 = 151.

Cf. A000041, A135010, A141285, A193870, A194446, A194447, A206437, A211009, A211978, A220517, A225600, A225610, A228109, A228110, A228350, A230440, A233968.

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.

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.

A211004 Number of distinct regions in the set of partitions of n.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 12, 15, 18, 22, 26, 30, 35, 40, 45, 51

Offset: 1

Omar E. Pol, Oct 22 2012

The number of regions in the set of partitions of n equals the number of partitions of n. The sequence counts only the distinct regions. For the definition of "regions of the set of partitions of n" (or more simply "regions of n") see A206437.

Is this the same as A001840 for all positive integers? If not, where is the first place these sequences differ?

			For n = 6 the 11 regions of 6 are [1], [2,1], [3,1,1], [2], [4,2,1,1,1], [3], [5,2,1,1,1,1,1], [2], [4,2], [3], [6,3,2,2,1,1,1,1,1,1,1]. These number are the first A006128(6) terms of triangle A206437 in which the first A000041(6) rows are the 11 regions of 6. We can see that the 8th region is equal to the 4th region: [2] = [2]. Also the 10th region is equal to the 6th region: [3] = [3]. There are two repeated regions, therefore a(6) = A000041(6) - 2 = 11 - 2 = 9.

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

Cf. A000041, A006128, A135010, A141285, A182244, A186114, A186412, A187219, A193870, A194446, A194447, A206437, A211009.

A225599 Triangle read by rows: T(n,k) = sum of all parts that start in the k-th column of the diagram of regions of the set of partitions of n.

Original entry on oeis.org

1, 3, 1, 6, 1, 2, 12, 1, 4, 3, 20, 1, 4, 5, 5, 35, 1, 6, 8, 9, 7, 54, 1, 6, 10, 12, 11, 11, 86, 1, 8, 13, 20, 14, 19, 15, 128, 1, 8, 18, 23, 22, 25, 23, 22, 192, 1, 10, 21, 34, 30, 37, 29, 36, 30, 275, 1, 10, 26, 41, 41, 48, 41, 45, 46, 42, 399, 1, 12, 32, 56, 53, 72, 52, 67, 58, 66, 56

Offset: 1

Omar E. Pol, Aug 02 2013

For the construction of the diagram see A225600.

			For n = 5 and k = 3 the diagram of regions of the set of partitions of 5 contains three parts that start in the third column: two parts of size 1 and one part of size 2, therefore the sum of all parts that start in column 3 is 1 + 1 + 2 = 4, so T(5,3) = 4.
.
.                       Illustration of the parts
.    Diagram             that start 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:  20           1    4      5      5
.
Triangle begins:
1;
3,   1;
6,   1,  2;
12,  1,  4,  3;
20,  1,  4,  5,  5;
35,  1,  6,  8,  9,  7;
54,  1,  6, 10, 12, 11, 11;
86,  1,  8, 13, 20, 14, 19, 15;
128, 1,  8, 18, 23, 22, 25, 23, 22;
192, 1, 10, 21, 34, 30, 37, 29, 36, 30;
275, 1, 10, 26, 41, 41, 48, 41, 45, 46, 42;
399, 1, 12, 32, 56, 53, 72, 52, 67, 58, 66, 56;

Column 1-2: A006128, A000012. Row sums give A066186. Right border gives A000041.

Cf. A135010, A141285, A176572, A186114, A186412, A187219, A193870, A194446, A206437, A211978, A225598, A225600, A225610.

cp's OEIS Frontend

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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A194799 Triangle read by rows: T(n,k) = number of partitions of n that are formed by k sections, k >= 1.

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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.

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A244968 Area between two valleys at height 0 under the infinite Dyck path related to partitions in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, k >= 1, multiplied by 2.

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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.

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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).

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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.

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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.

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A211004 Number of distinct regions in the set of partitions of n.

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A225599 Triangle read by rows: T(n,k) = sum of all parts that start in the k-th column of the diagram of regions of the set of partitions of n.

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