A141285 -id:A141285 - cp's OEIS frontend

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

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.

A212125 Largest part of n-th partition of the table of partitions of A211999.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 14 2012

nonn

Cf. A135010, A141285, A211999.

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.

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.

A256010 Product of n and the total number of parts in all partitions of n. Also, product of n and the sum of largest parts of all partitions of n.

Original entry on oeis.org

0, 1, 6, 18, 48, 100, 210, 378, 688, 1152, 1920, 3025, 4788, 7228, 10920, 16020, 23408, 33405, 47592, 66462, 92600, 127092, 173778, 234738, 316176, 421275, 559572, 736938, 967260, 1260137, 1636890, 2112185, 2717664, 3477078, 4435708, 5630660, 7128504, 8984044, 11293638, 14140893, 17661840, 21980264, 27291222

Offset: 0

Omar E. Pol, May 31 2015

nonn

a(n) is also the volume of a three-dimensional model of partitions which is a polycube puzzle that contains n sections and A000041(n) pieces related to the A000041(n) regions of the set of partitions of n. The volume is equivalent to a(n) unit cubes.

			For n = 6 the total number of parts in all partitions of 6 is equal to 35 so a(n) = 6 * 35 = 210. On the other hand, the sum of largest parts of all partitions of 6 is 1 + 2 + 3 + 2 + 4 + 3 + 5 + 2 + 4 + 3 + 6 = 35, so a(6) is also 6 * 35 = 210.
Illustration of three views of a three-dimensional model of partitions after 6th stage:
.
.                     y
.
.                  _  |  _ _ _ _ _ _
.                _|_| | |_ _ _      |
.               | |_| | |_ _ _|_    |
.              _|_|_| | |_ _    |   |
.             |_|_|_| | |_ _|_ _|_  |
.              _|_|_| | |_ _ _    | |
.             | | |_| | |_ _ _|_  | |
.            _|_|_|_| | |_ _    | | |
.           | | | |_| | |_ _|_  | | |
.          _|_|_|_|_| | |_ _  | | | |
.        _|_|_|_|_|_| | |_  | | | | |
.       |_|_|_|_|_|_| | |_|_|_|_|_|_|
.   z  _ _ _ _ _ _ _ _|_ _ _ _ _ _ _ _  x
.                     |  _ _ _ _ _ _
.                     | |_| | | | | |
.                     | |_ _| | | | |
.                     | |_ _ _| | | |
.                     | |_ _ _ _| | |
.                     | |_ _ _ _ _| |
.                     | |_ _ _ _ _ _|
.                     |
.
.                     z
.
For n = 6 the areas of the views are A006128(6) = 35, A066186(6) = 6 * 11 = 66 and A000290(6) = 6^2 = 36. The structure contains A000041(6) = 11 pieces and the volume is equal to a(6) = 6 * 35 = 210.

Cf. A000041, A000290, A006128, A066186, A135010, A141285, A194446, A206437.

lim = 42; CoefficientList[Series[Sum[n x^n Product[1/(1 - x^k), {k, n}], {n, lim}], {x, 0, lim}], x] Range[0, lim] (* Michael De Vlieger, Jul 14 2015, after N. J. A. Sloane at A006128 *)

a(n) = n * A006128(n).

A330242 Sum of largest emergent parts of the partitions of n.

Original entry on oeis.org

0, 0, 0, 2, 3, 9, 12, 24, 33, 54, 72, 112, 144, 210, 273, 379, 485, 661, 835, 1112, 1401, 1825, 2284, 2944, 3652, 4645, 5745, 7223, 8879, 11080, 13541, 16760, 20406, 25062, 30379, 37102, 44761, 54351, 65347, 78919, 94517, 113645, 135603, 162331, 193088, 230182, 272916, 324195, 383169, 453571

Offset: 1

Omar E. Pol, Dec 06 2019

nonn

In other words: a(n) is the sum of the largest parts of all partitions of n that contain emergent parts.
The partitions of n that contain emergent parts are the partitions that contain neither 1 nor n as a part. All parts of these partitions are emergent parts except the last part of every partition.
For the definition of emergent part see A182699.

			For n = 9 the diagram of
the partitions of 9 that
do not contain 1 as a part
is as shown below:           Partitions
.
    |_ _ _|   |   |   |      [3, 2, 2, 2]
    |_ _ _ _ _|   |   |      [5, 2, 2]
    |_ _ _ _|     |   |      [4, 3, 2]
    |_ _ _ _ _ _ _|   |      [7, 2]
    |_ _ _|     |     |      [3, 3, 3]
    |_ _ _ _ _ _|     |      [6, 3]
    |_ _ _ _ _|       |      [5, 4]
    |_ _ _ _ _ _ _ _ _|      [9]
.
Note that the above diagram is also the "head" of the last section of the set of partitions of 9, where the "tail" is formed by A000041(9-1)= 22 1's.
The diagram of the
emergent parts is as
shown below:                 Emergent parts
.
    |_ _ _|   |   |          [3, 2, 2]
    |_ _ _ _ _|   |          [5, 2]
    |_ _ _ _|     |          [4, 3]
    |_ _ _ _ _ _ _|          [7]
    |_ _ _|     |            [3, 3]
    |_ _ _ _ _ _|            [6]
    |_ _ _ _ _|              [5]
.
The sum of the largest emergent parts is 3 + 5 + 4 + 7 + 3 + 6 + 5 = 33, so a(9) = 33.

Cf. A000041, A002865, A006128, A135010, A138135, A138137, A141285, A182699, A182703, A182709, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A207031, A299474, A299475.

a(n) = A138137(n) - n.

a(n) = A207031(n,1) - n.

cp's OEIS Frontend

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

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A212125 Largest part of n-th partition of the table of partitions of A211999.

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A220487 Partial sums of triangle A206437.

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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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A256010 Product of n and the total number of parts in all partitions of n. Also, product of n and the sum of largest parts of all partitions of n.

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A330242 Sum of largest emergent parts of the partitions of n.

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