A194446 -id:A194446 - cp's OEIS frontend

A278355 a(n) = sum of the perimeters of the Ferrers boards of the partitions of n. Also, sum of the perimeters of the diagrams of the regions of the set of partitions of n.

0, 4, 12, 24, 48, 80, 140, 216, 344, 512, 768, 1100, 1596, 2224, 3120, 4272, 5852, 7860, 10576, 13992, 18520, 24208, 31596, 40824, 52696, 67404, 86088, 109176, 138180, 173812, 218252, 272540, 339708, 421464, 521848, 643504, 792056, 971248, 1188804, 1450348, 1766184, 2144416, 2599164, 3141748, 3791248, 4563780

Offset: 0

Omar E. Pol, Nov 19 2016

nonn

a(n) is also 4 times the total number of parts in all partitions of n.
Hence a(n) is also 4 times the sum of largest parts of all partitions of n.
Hence a(n) is also twice the total number of parts in all partitions of n plus twice the sum of largest parts of all partitions of n.
a(n) is also the sum of the perimeters of the first n polygons constructed with the Dyck path (and its mirror) that arises from the minimalist diagram of the regions of the set of partitions of n. The n-th odd-indexed segment of the diagram has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. The k-th polygon of the diagram is associated to the k-th section of the set of partitions of n, with 1<=k<=n. See the bottom of Example section. For the definition of "section" see A135010. For the definition of "region" see A206437.

			For n = 5 consider the partitions of 5 in colexicographic order (as shown in the 5th row of the triangle A211992) and its associated diagram of regions as shown below:
.                                Regions            Minimalist
.         Partitions of 5        diagram             version
.                               _ _ _ _ _
.         1, 1, 1, 1, 1        |_| | | | |          _| | | | |
.         2, 1, 1, 1           |_ _| | | |          _ _| | | |
.         3, 1, 1              |_ _ _| | |          _ _ _| | |
.         2, 2, 1              |_ _|   | |          _ _|   | |
.         4, 1                 |_ _ _ _| |          _ _ _ _| |
.         3, 2                 |_ _ _|   |          _ _|     |
.         5                    |_ _ _ _ _|          _ _ _ _ _|
.
Then consider the following table which contains the Ferrers boards of the partitions of 5 and the diagram of every region of the set of partitions of 5:
-------------------------------------------------------------------------
| Partitions  |             |       |   Regions   |             |       |
|     of 5    |   Ferrers   | Peri- |     of 5    |   Region    | Peri- |
|(See A211992)|    board    | meter |(see A220482)|   diagram   | meter |
-------------------------------------------------------------------------
|                  _                |                 _                 |
|      1          |_|               |       1        |_|            4   |
|      1          |_|               |                   _               |
|      1          |_|               |       1         _|_|              |
|      1          |_|               |       2        |_|_|          8   |
|      1          |_|          12   |                     _             |
|                  _ _              |       1            |_|            |
|      2          |_|_|             |       1         _ _|_|            |
|      1          |_|               |       3        |_|_|_|       12   |
|      1          |_|               |                 _ _               |
|      1          |_|          12   |       2        |_|_|          6   |
|                  _ _ _            |                       _           |
|      3          |_|_|_|           |       1              |_|          |
|      1          |_|               |       1              |_|          |
|      1          |_|          12   |       1             _|_|          |
|                  _ _              |       2         _ _|_|_|          |
|      2          |_|_|             |       4        |_|_|_|_|     18   |
|      2          |_|_|             |                 _ _ _             |
|      1          |_|          10   |       3        |_|_|_|        8   |
|                  _ _ _ _          |                         _         |
|      4          |_|_|_|_|         |       1                |_|        |
|      1          |_|          12   |       1                |_|        |
|                  _ _ _            |       1                |_|        |
|      3          |_|_|_|           |       1                |_|        |
|      2          |_|_|        10   |       1               _|_|        |
|                  _ _ _ _ _        |       2         _ _ _|_|_|        |
|      6          |_|_|_|_|_|  12   |       5        |_|_|_|_|_|   24   |
|                                   |                                   |
-------------------------------------------------------------------------
|   Sum of perimeters:         80         <-- equals -->           80   |
-------------------------------------------------------------------------
The sum of the perimeters of the Ferrers boards is 12 + 12 + 12 + 10 + 12 + 10 + 12 = 80, so a(5) = 80.
On the other hand, the sum of the perimeters of the diagrams of regions is 4 + 8 + 12 + 6 + 18 + 8 + 24 = 80, equaling the sum of the perimeters of the Ferrers boards.
.
Illustration of first six polygons of an infinite diagram constructed with the boundary segments of the minimalist diagram of regions and its mirror (note that the diagram looks like reflections on a mountain lake):
11............................................................
.                                                            /\
.                                                           /  \
.                                                          /    \
7...................................                      /      \
.                                  /\                    /        \
5.....................            /  \                /\/          \
.                    /\          /    \          /\  /              \
3...........        /  \        /      \        /  \/                \
2.......   /\      /    \    /\/        \      /                      \
1...  /\  /  \  /\/      \  /            \  /\/                        \
0  /\/  \/    \/          \/              \/                            \
.  \/\  /\    /\          /\              /\                            /
.     \/  \  /  \/\      /  \            /  \/\                        /
.          \/      \    /    \/\        /      \                      /
.                   \  /        \      /        \  /\                /
.                    \/          \    /          \/  \              /
.                                 \  /                \/\          /
.                                  \/                    \        /
.                                                         \      /
.                                                          \    /
.                                                           \  /
.                                                            \/
n:
. 0 1  2   3          4             5                         6
Perimeter of the n-th polygon:
. 0 4  8  12         24            32                        60
a(n) is the sum of the perimeters of the first n polygons:
. 0 4 12  24         48            80                       140
.
For n = 5, the sum of the perimeters of the first five polygons is 4 + 8 + 12 + 24 + 32 = 80, so a(5) = 80.
For n = 6, the sum of the perimeters of the first six polygons is 4 + 8 + 12 + 24 + 32 + 60 = 140, so a(6) = 140.
For another version of the above diagram see A228109.

Cf. A000041, A006128, A135010, A138137, A139582, A141285, A194446, A211992, A220482, A225600, A211978, A233968, A244968.

a(n) = 4*A006128(n) = 2*A211978(n).
a(n) = 2*A225600(2*A000041(n)) = 2*A225600(A139582(n)), n >= 1.
a(n) = 2*((Sum_{m=1..p(n)} A194446(m)) + (Sum_{m=1..p(n)} A141285(m))) = 4*Sum_{m=1..p(n)} A194446(m) = 4*Sum_{m=1..p(n)} A141285(m), where p(n) = A000041(n), n >= 1.

A299474 a(n) = 4*p(n), where p(n) is the number of partitions of n.

Original entry on oeis.org

4, 4, 8, 12, 20, 28, 44, 60, 88, 120, 168, 224, 308, 404, 540, 704, 924, 1188, 1540, 1960, 2508, 3168, 4008, 5020, 6300, 7832, 9744, 12040, 14872, 18260, 22416, 27368, 33396, 40572, 49240, 59532, 71908, 86548, 104060, 124740, 149352, 178332, 212696, 253044, 300700, 356536, 422232, 499016, 589092, 694100, 816904

Offset: 0

Omar E. Pol, Feb 10 2018

nonn

For n >= 1, a(n) is also the number of edges in the diagram of partitions of n, in which A299475(n) is the number of vertices and A000041(n) is the number of regions (see example and Euler's formula).

			Construction of a modular table of partitions in which a(n) is the number of edges of the diagram after n-th stage (n = 1..6):
--------------------------------------------------------------------------------
n ........:   1     2       3         4           5             6     (stage)
a(n)......:   4     8      12        20          28            44     (edges)
A299475(n):   4     7      10        16          22            34     (vertices)
A000041(n):   1     2       3         5           7            11     (regions)
--------------------------------------------------------------------------------
r     p(n)
--------------------------------------------------------------------------------
.             _    _ _    _ _ _    _ _ _ _    _ _ _ _ _    _ _ _ _ _ _
1 .... 1 ....|_|  |_| |  |_| | |  |_| | | |  |_| | | | |  |_| | | | | |
2 .... 2 .........|_ _|  |_ _| |  |_ _| | |  |_ _| | | |  |_ _| | | | |
3 .... 3 ................|_ _ _|  |_ _ _| |  |_ _ _| | |  |_ _ _| | | |
4                                 |_ _|   |  |_ _|   | |  |_ _|   | | |
5 .... 5 .........................|_ _ _ _|  |_ _ _ _| |  |_ _ _ _| | |
6                                            |_ _ _|   |  |_ _ _|   | |
7 .... 7 ....................................|_ _ _ _ _|  |_ _ _ _ _| |
8                                                         |_ _|   |   |
9                                                         |_ _ _ _|   |
10                                                        |_ _ _|     |
11 .. 11 .................................................|_ _ _ _ _ _|
.
Apart from the axis x, the r-th horizontal line segment has length A141285(r), equaling the largest part of the r-th region of the diagram.
Apart from the axis y, the r-th vertical line segment has length A194446(r), equaling the number of parts in the r-th region of the diagram.
The total number of parts equals the sum of largest parts.
Note that every diagram contains all previous diagrams.
An infinite diagram is a table of all partitions of all positive integers.

Shawn A. Broyles, Table of n, a(n) for n = 0..1000

k times partition numbers: A000041 (k=1), A139582 (k=2), A299473 (k=3), this sequence (k=4).

Cf. A135010, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299475.

List([0..50],n->4*NrPartitions(n)); # Muniru A Asiru, Jul 10 2018

with(combinat): seq(4*numbpart(n),n=0..50); # Muniru A Asiru, Jul 10 2018

4*PartitionsP[Range[0,50]] (* Harvey P. Dale, Dec 05 2023 *)

a(n) = 4*numbpart(n); \\ Michel Marcus, Jul 15 2018

from sympy.ntheory import npartitions
def a(n): return 4*npartitions(n)
print([a(n) for n in range(51)]) # Michael S. Branicky, Apr 04 2021

a(n) = 4*A000041(n) = 2*A139582(n).
a(n) = A000041(n) + A299475(n) - 1, n >= 1 (Euler's formula).
a(n) = A000041(n) + A299473(n). - Omar E. Pol, Aug 11 2018

A299475 a(n) is the number of vertices in the diagram of partitions of n (see example).

Original entry on oeis.org

1, 4, 7, 10, 16, 22, 34, 46, 67, 91, 127, 169, 232, 304, 406, 529, 694, 892, 1156, 1471, 1882, 2377, 3007, 3766, 4726, 5875, 7309, 9031, 11155, 13696, 16813, 20527, 25048, 30430, 36931, 44650, 53932, 64912, 78046, 93556, 112015, 133750, 159523, 189784, 225526, 267403, 316675, 374263, 441820, 520576, 612679

Offset: 0

Omar E. Pol, Feb 11 2018

nonn

For n >= 1, A299474(n) is the number of edges and A000041(n) is the number of regions in the mentioned diagram (see example and Euler's formula).

			Construction of a modular table of partitions in which a(n) is the number of vertices of the diagram after n-th stage (n = 1..6):
--------------------------------------------------------------------------------
n ........:   1     2       3         4           5             6     (stage)
a(n)......:   4     7      10        16          22            34     (vertices)
A299474(n):   4     8      12        20          28            44     (edges)
A000041(n):   1     2       3         5           7            11     (regions)
--------------------------------------------------------------------------------
r     p(n)
--------------------------------------------------------------------------------
.             _    _ _    _ _ _    _ _ _ _    _ _ _ _ _    _ _ _ _ _ _
1 .... 1 ....|_|  |_| |  |_| | |  |_| | | |  |_| | | | |  |_| | | | | |
2 .... 2 .........|_ _|  |_ _| |  |_ _| | |  |_ _| | | |  |_ _| | | | |
3 .... 3 ................|_ _ _|  |_ _ _| |  |_ _ _| | |  |_ _ _| | | |
4                                 |_ _|   |  |_ _|   | |  |_ _|   | | |
5 .... 5 .........................|_ _ _ _|  |_ _ _ _| |  |_ _ _ _| | |
6                                            |_ _ _|   |  |_ _ _|   | |
7 .... 7 ....................................|_ _ _ _ _|  |_ _ _ _ _| |
8                                                         |_ _|   |   |
9                                                         |_ _ _ _|   |
10                                                        |_ _ _|     |
11 .. 11 .................................................|_ _ _ _ _ _|
.
Apart from the axis x, the r-th horizontal line segment has length A141285(r), equaling the largest part of the r-th region of the diagram.
Apart from the axis y, the r-th vertical line segment has length A194446(r), equaling the number of parts in the r-th region of the diagram.
The total number of parts equals the sum of largest parts.
Note that every diagram contains all previous diagrams.
An infinite diagram is a table of all partitions of all positive integers.

Shawn A. Broyles, Table of n, a(n) for n = 0..1000

Cf. A000041, A135010, A139582, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299474.

a(n) = if (n==0, 1, 1+3*numbpart(n)); \\ Michel Marcus, Jul 15 2018

a(0) = 1; a(n) = 1 + 3*A000041(n), n >= 1.

a(n) = A299474(n) - A000041(n) + 1, n >= 1 (Euler's formula).

A179862 An unrestricted partition statistic: sum of A179864 over row n.

Original entry on oeis.org

1, 4, 9, 19, 33, 59, 93, 150, 226, 342, 494, 721, 1011, 1425, 1960, 2695, 3633, 4903, 6506, 8633, 11312, 14796, 19157, 24773, 31744, 40608, 51578, 65372, 82341, 103522, 129428, 161505, 200589, 248614, 306869, 378051, 463987, 568387, 693989, 845754, 1027625

Offset: 1

Alford Arnold, Aug 02 2010

nonn

Total number of parts in all partitions of n plus the sum of largest parts of all partitions of n minus the number of partitions of n. - Omar E. Pol, Jul 15 2013

Sum of the hook-lengths of the (1,1)-cells of the Ferrers diagrams over all partitions of n. Example: a(3) = 9 because in each of the partitions 3, 21, and 111 the (1,1)-cell has hook-length 3. Comment follows at once from the previous comment. - Emeric Deutsch, Dec 20 2015

			From _Omar E. Pol_, Jul 15 2013: (Start)
Illustration of initial terms using a Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). a(n) is the x-coordinate of the mentioned largest peak. Note that this Dyck path is infinite.
.
7..................................
.                                 /\
5....................            /  \                /\
.                   /\          /    \          /\  /
3..........        /  \        /      \        /  \/
2.....    /\      /    \    /\/        \      /
1..  /\  /  \  /\/      \  /            \  /\/
0 /\/  \/    \/          \/              \/
. 0,2,  6,   12,         24,             40... = A211978
.  1, 4,   9,       19,           33... = this sequence (End)

Cf. A179864.

a(n) = Sum_{k=1..A000041(n)} A179864(n,k).
a(n) = A211978(n) - A000041(n). - Omar E. Pol, Jul 15 2013
a(n) = A225600(A139582(n)-1), n>= 1. - Omar E. Pol, Jul 25 2013

More terms from Omar E. Pol, Jul 15 2013

A299773 a(n) is the index of the partition that contains the divisors of n in the list of colexicographically ordered partitions of the sum of the divisors of n.

Original entry on oeis.org

1, 2, 3, 9, 7, 48, 15, 119, 72, 269, 56, 2740, 101, 1163, 1208, 5218, 297, 24319, 490, 42150, 6669, 14098, 1255, 792335, 5564, 42501, 30585, 432413, 4565, 4513067, 6842, 1251217, 122818, 317297, 124253, 54782479, 21637, 802541, 445414, 48590725, 44583

Offset: 1

Omar E. Pol, Mar 25 2018

nonn

If n is a noncomposite number (that is, 1 or prime), then a(n) = A000041(n).

For n >= 3, p(sigma(n-2)) < a(n) <= p(sigma(n-1)), where p(n) = A000041(n) and sigma(n) = A000203(n).

			For n = 4 the sum of the divisors of 4 is 1 + 2 + 4 = 7. Then we have that, in list of colexicographically ordered partitions of 7, the divisors of 4 are in the 9th partition, so a(4) = 9 (see below):
------------------------------------------------------
   k        Diagram        Partitions of 7
------------------------------------------------------
         _ _ _ _ _ _ _
   1    |_| | | | | | |    [1, 1, 1, 1, 1, 1, 1]
   2    |_ _| | | | | |    [2, 1, 1, 1, 1, 1]
   3    |_ _ _| | | | |    [3, 1, 1, 1, 1]
   4    |_ _|   | | | |    [2, 2, 1, 1, 1]
   5    |_ _ _ _| | | |    [4, 1, 1, 1]
   6    |_ _ _|   | | |    [3, 2, 1, 1]
   7    |_ _ _ _ _| | |    [5, 1, 1]
   8    |_ _|   |   | |    [2, 2, 2, 1]
   9    |_ _ _ _|   | |    [4, 2, 1]       <---- Divisors of 4
  10    |_ _ _|     | |    [3, 3, 1]
  11    |_ _ _ _ _ _| |    [6, 1]
  12    |_ _ _|   |   |    [3, 2, 2]
  13    |_ _ _ _ _|   |    [5, 2]
  14    |_ _ _ _|     |    [4, 3]
  15    |_ _ _ _ _ _ _|    [7]
.

Amiram Eldar, Table of n, a(n) for n = 1..3200 (terms 1..700 from Andrew Howroyd)

Cf. A000040, A000041, A000203, A008578, A026807, A027750, A056538, A135010, A141285, A194446, A211992, A272024.

b[n_, k_] := b[n, k] = If[k < 1 || k > n, 0, If[n == k, 1, b[n, k + 1] + b[n - k, k]]];
PartIndex[v_] := Module[{s = 1, t = 0}, For[i = Length[v], i >= 1, i--, t += v[[i]]; s += b[t, If[i == 1, 1, v[[i - 1]]]] - b[t, v[[i]]]]; s];
a[n_] := PartIndex[Divisors[n]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 27 2019, after Andrew Howroyd *)

a(n)={my(d=divisors(n)); vecsearch(vecsort(partitions(vecsum(d))), d)} \\ Andrew Howroyd, Jul 15 2018

\\ here b(n,k) is A026807.
b(n,k)=polcoeff(1/prod(i=k, n, 1-x^i + O(x*x^n)), n)
PartIndex(v)={my(s=1,t=0); forstep(i=#v, 1, -1, t+=v[i]; s+=b(t, if(i==1, 1, v[i-1])) - b(t, v[i])); s}
a(n)=PartIndex(divisors(n)); \\ Andrew Howroyd, Jul 15 2018

Terms a(8) and beyond from Andrew Howroyd, Jul 15 2018

A273140 Number of parts in the corner of size n X n of the modular table of partitions described in Comments.

Original entry on oeis.org

1, 3, 6, 11, 17, 25, 34, 46, 59, 74, 90, 109, 129, 151, 174, 201, 229, 259, 290, 323, 358, 394, 434, 475, 518, 562, 609, 657, 707, 758, 814, 871, 930, 990, 1052, 1116, 1181, 1249, 1318, 1389, 1462, 1536, 1615, 1695, 1777, 1860, 1946, 2033, 2122, 2212, 2305, 2400, 2496, 2594, 2694, 2795

Offset: 1

Omar E. Pol, May 16 2016

nonn

Consider an infinite dissection of the fourth quadrant of the square grid in which, apart from the axes x and y, the k-th horizontal line segment has length A141285(n) and the n-th vertical line segment has length A194446(n). Both line segments shares the point (A141285(n),n). Note that in the infinite table there are no partitions because every row contains an infinite number of parts. On the other hand, taking only the first k sections from the table we have a representation of the partitions of k. For the definition of "region" see A206437. For the definition of "section" see A135010.

			For n = 4 the corner of size 4 X 4 of the modular table of partitions contains 11 parts as shown below, so a(4) = 11.
.
.   Row   _ _ _ _       Parts
.    1   |_| | | |        4
.    2   |_ _| | |        3
.    3   |_ _ _| |        2
.    4   |_ _|   |        2
.                       ----
.                  Total 11
.
For n = 20 the corner of size 20 X 20 of the modular table of partitions contains 323 parts as shown below, so a(20) = 323.
.
.   Row   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _       Parts
.    1   |_| | | | | | | | | | | | | | | | | | | |        20
.    2   |_ _| | | | | | | | | | | | | | | | | | |        19
.    3   |_ _ _| | | | | | | | | | | | | | | | | |        18
.    4   |_ _|   | | | | | | | | | | | | | | | | |        18
.    5   |_ _ _ _| | | | | | | | | | | | | | | | |        17
.    6   |_ _ _|   | | | | | | | | | | | | | | | |        17
.    7   |_ _ _ _ _| | | | | | | | | | | | | | | |        16
.    8   |_ _|   |   | | | | | | | | | | | | | | |        17
.    9   |_ _ _ _|   | | | | | | | | | | | | | | |        16
.   10   |_ _ _|     | | | | | | | | | | | | | | |        16
.   11   |_ _ _ _ _ _| | | | | | | | | | | | | | |        15
.   12   |_ _ _|   |   | | | | | | | | | | | | | |        16
.   13   |_ _ _ _ _|   | | | | | | | | | | | | | |        15
.   14   |_ _ _ _|     | | | | | | | | | | | | | |        15
.   15   |_ _ _ _ _ _ _| | | | | | | | | | | | | |        14
.   16   |_ _|   |   |   | | | | | | | | | | | | |        16
.   17   |_ _ _ _|   |   | | | | | | | | | | | | |        15
.   18   |_ _ _|     |   | | | | | | | | | | | | |        15
.   19   |_ _ _ _ _ _|   | | | | | | | | | | | | |        14
.   20   |_ _ _ _ _|     | | | | | | | | | | | | |        14
.                                                       -----
.                                                  Total 323
.

Cf. A000041, A006128, A135010, A141285, A194446, A206437, A211992, A220482, A228525, A278602.

A278602 Sum of the perimeters of all regions of the n-th section of a modular table of partitions.

Original entry on oeis.org

0, 4, 8, 12, 24, 32, 60, 76, 128, 168, 256, 332, 496, 628, 896, 1152, 1580, 2008, 2716, 3416, 4528, 5688, 7388, 9228, 11872, 14708, 18684, 23088, 29004, 35632, 44440, 54288, 67168, 81756, 100384, 121656, 148552, 179192, 217556, 261544, 315836, 378232, 454748, 542584, 649500, 772532, 920912

Offset: 0

Omar E. Pol, Nov 23 2016

nonn

Consider an infinite dissection of the fourth quadrant of the square grid in which, apart from the axes x and y, the k-th horizontal line segment has length A141285(k) and the k-th vertical line segment has length A194446(k). Both line segments shares the point (A141285(k),k). For n>=1, the table contains A000041(n) regions which are distributed in n sections. Note that in the infinite table there are no partitions because every row contains an infinite number of parts. On the other hand, taking only the first n sections from the table we have a representation of the partitions of n. For an illustration see the example. For the definition of "region" see A206437. For the definition of "section" see A135010. For a visualization of the corner of size n X n of the table see A273140.

a(n) is also the sum of the perimeters of the Ferrers boards of the partitions of n, minus the sum of the perimeters of the Ferrers boards of the partitions of n-1, with n >= 1. For more information see A278355.

			For n = 1..6, consider the modular table of partitions for the first six positive integers as shown below in the fourth quadrant of the square grid (see Figure 1):
|--------------|-----------------------------------------------------|
| Modular table|                      Sections                       |
| of partitions|-----------------------------------------------------|
|  for n=1..6  | 1     2       3         4           5             6 |
1--------------|-----------------------------------------------------|
.  _ _ _ _ _ _   _     _       _         _           _             _
. |_| | | | | | |_|  _| |     | |       | |         | |           | |
. |_ _| | | | |     |_ _|  _ _| |       | |         | |           | |
. |_ _ _| | | |           |_ _ _|  _ _ _| |         | |           | |
. |_ _|   | | |                   |_ _|   |         | |           | |
. |_ _ _ _| | |                   |_ _ _ _|  _ _ _ _| |           | |
. |_ _ _|   | |                             |_ _ _|   |           | |
. |_ _ _ _ _| |                             |_ _ _ _ _|  _ _ _ _ _| |
. |_ _|   |   |                                         |_ _|   |   |
. |_ _ _ _|   |                                         |_ _ _ _|   |
. |_ _ _|     |                                         |_ _ _|     |
. |_ _ _ _ _ _|                                         |_ _ _ _ _ _|
.
.   Figure 1.                         Figure 2.
.
The table contains 11 regions, see Figure 1.
The regions are distributed in 6 sections. The Figure 2 shows the sections separately.
Then consider the following table which contains the diagram of every region separately:
---------------------------------------------------------------------
|         |         |         |                    |       |        |
| Section | Region  |  Parts  |       Region       | Peri- |  a(n)  |
|         |         |(A220482)|       diagram      | meter |        |
---------------------------------------------------------------------
|         |         |         |      _             |       |        |
|    1    |    1    |    1    |     |_|            |   4   |    4   |
---------------------------------------------------------------------
|         |         |         |        _           |       |        |
|         |         |    1    |      _| |          |       |        |
|    2    |    2    |    2    |     |_ _|          |   8   |    8   |
---------------------------------------------------------------------
|         |         |         |          _         |       |        |
|         |         |    1    |         | |        |       |        |
|         |         |    1    |      _ _| |        |       |        |
|    3    |    3    |    3    |     |_ _ _|        |  12   |   12   |
---------------------------------------------------------------------
|         |         |         |      _ _           |       |        |
|         |    4    |    2    |     |_ _|          |   6   |        |
|         |---------|---------|----------------------------|        |
|         |         |         |            _       |       |        |
|         |         |    1    |           | |      |       |        |
|         |         |    1    |           | |      |       |        |
|         |         |    1    |          _| |      |       |        |
|         |         |    2    |      _ _|   |      |       |        |
|    4    |    5    |    4    |     |_ _ _ _|      |  18   |   24   |
---------------------------------------------------------------------
|         |         |         |      _ _ _         |       |        |
|         |    6    |    3    |     |_ _ _|        |   8   |        |
|         |---------|---------|--------------------|-------|        |
|         |         |         |              _     |       |        |
|         |         |    1    |             | |    |       |        |
|         |         |    1    |             | |    |       |        |
|         |         |    1    |             | |    |       |        |
|         |         |    1    |             | |    |       |        |
|         |         |    1    |            _| |    |       |        |
|         |         |    2    |      _ _ _|   |    |       |        |
|    5    |    7    |    5    |     |_ _ _ _ _|    |  24   |   32   |
---------------------------------------------------------------------
|         |         |         |      _ _           |       |        |
|         |    8    |    2    |     |_ _|          |   6   |        |
|         |---------|---------|--------------------|-------|        |
|         |         |         |          _ _       |       |        |
|         |         |    2    |      _ _|   |      |       |        |
|         |    9    |    4    |     |_ _ _ _|      |  12   |        |
1         |---------|---------|--------------------|-------|        |
|         |         |         |      _ _ _         |       |        |
|         |   10    |    3    |     |_ _ _|        |   8   |        |
|         |---------|---------|--------------------|-------|        |
|         |         |         |                _   |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |               | |  |       |        |
|         |         |    1    |              _| |  |       |        |
|         |         |    2    |             |   |  |       |        |
|         |         |    2    |            _|   |  |       |        |
|         |         |    3    |      _ _ _|     |  |       |        |
|    6    |   11    |    6    |     |_ _ _ _ _ _|  |  34   |   60   |
---------------------------------------------------------------------
.
For n = 1..3, there is only one region in every section. The perimeters of the regions are 4, 8 and 12 respectively, so a(1) = 4, a(2) = 8, and a(3) = 12.
For n = 4, the 4th section contains two regions with perimeters 6 and 18 respectively. The sum of the perimeters is 6 + 18 = 24, so a(4) = 24.
For n = 5, the 5th section contains two regions with perimeters 8 and 24 respectively. The sum of the perimeters is 8 + 24 = 32, so a(5) = 32.
For n = 6, the 6th section contains four regions with perimeters 6, 12, 8 and 34 respectively. The sum of the perimeters is 6 + 12 + 8 + 34 = 60, so a(6) = 60.

Partial sums give A278355.

Cf. A000041, A135010, A138137, A141285, A194446, A206437, A211992, A220482, A233968, A273140.

a(n) = 4 * A138137(n) = 2 * A233968(n), n >= 1 in both cases.

A210969 Sum of all region numbers of all parts of the last section of the set of partitions of n.

Original entry on oeis.org

1, 4, 9, 29, 55, 157, 277, 669, 1212, 2555, 4459, 9048

Offset: 1

Omar E. Pol, Jul 01 2012

Each part of a partition of n belongs to a different region of n. The "region number" of a part of the r-th region of n is equal to r. For the definition of "region of n" see A206437.

The last section of the set of partitions of n is also the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Apr 07 2014

			For n = 6 the four regions of the last section of 6 are [2], [4, 2], [3], [6, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1] therefore the "region numbers" are [8], [9, 9], [10], [11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11]. The sum of all region numbers is a(6) = 8+2*9+10+11^2 =  8+18+10+121 = 157, see below:
--------------------------------------------
.     Last section                  Sum of
.     of the set of     Region      region
k    partitions of 6    numbers     numbers
--------------------------------------------
11           6              11         11
10         3+3           10,11         21
9        4  +2         9,   11         20
8      2+2  +2       8,9,   11         28
7            1              11         11
6            1              11         11
5            1              11         11
4            1              11         11
3            1              11         11
2            1              11         11
1            1              11         11
--------------------------------------------
Total sum of region numbers is a(6) = 157

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

Row sums of triangle A210966. Partial sums give A210972.

Cf. A135010, A138121, A194446, A182703, A206437, A210971.

A210971 Triangle read by rows in which row n lists the region number of the parts of the k-th partition of n, with partitions reverse lexicographically ordered.

Original entry on oeis.org

1, 3, 2, 6, 5, 3, 11, 10, 8, 9, 5, 18, 17, 15, 16, 12, 13, 7, 29, 28, 26, 27, 23, 24, 18, 28, 20, 21, 11

Offset: 1

Omar E. Pol, Jun 30 2012

Each part of a partition of n belongs to a different region of n. The "region number" of a part of the r-th region of n is equal to r. For the definition of "region of n" see A206437.

			For n = 5 we have:
------------------------------------------------------
.              Two arrangements             Sum of
k           of the partitions of 5        partition k
------------------------------------------------------
7      [5]                          [5]        5
6      [3+2]                      [3+2]        5
5      [4+1]                    [4  +1]        5
4      [2+1+1]                [2+2  +1]        5
3      [3+1+1]              [3  +1  +1]        5
2      [2+1+1+1]          [2+1  +1  +1]        5
1      [1+1+1+1+1]      [1+1+1  +1  +1]        5
------------------------------------------------------
.              Two arrangements
.           of the region numbers           Sum of
k           of the partitions of 5          zone k
------------------------------------------------------
7      [7]                          [7]        7
6      [6,7]                      [6,7]       13
5      [5,7]                    [5,  7]       12
4      [4,5,7]                [4,5,  7]       16
3      [3,5,7]              [3,  5,  7]       15
2      [2,3,5,7]          [2,3,  5,  7]       17
1      [1,2,3,5,7]      [1,2,3,  5,  7]       18
------------------------------------------------------
So row 5 of triangle gives: 18, 17, 15, 16, 12, 13, 7.
.
Triangle begins:
1;
3,2;
6,5,3;
11,10,8,9,5;
18,17,15,16,12,13,7;
29,28,26,27,23,24,18,28,20,21,11;

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

Column 1 is A026905. Right border = row lengths = A000041, n>=1. Row sums give A210972.

Cf. A135010, A138121, A182703, A194446, A206437, A210969.

A210972 Sum of all region numbers of all parts of all partitions of n.

Original entry on oeis.org

1, 5, 14, 43, 98, 255, 532, 1201, 2413, 4968, 9427, 18475

Offset: 1

Omar E. Pol, Jun 30 2012

Each part of a partition of n belongs to a different region of n. The "region number" of a part of the r-th region of n is equal to r. For the definition of "region of n" see A206437.

			For n = 5 we have:
---------------------------------------------------
.              Two arrangements
k           of the partitions of 5
---------------------------------------------------
7      [5]                          [5]
6      [3+2]                      [3+2]
5      [4+1]                    [4  +1]
4      [2+1+1]                [2+2  +1]
3      [3+1+1]              [3  +1  +1]
2      [2+1+1+1]          [2+1  +1  +1]
1      [1+1+1+1+1]      [1+1+1  +1  +1]
---------------------------------------------------
.              Two arrangements
.           of the region numbers           Sum of
k           of the partitions of 5          zone k
---------------------------------------------------
7      [7]                          [7]        7
6      [6,7]                      [6,7]       13
5      [5,7]                    [5,  7]       12
4      [4,5,7]                [4,5,  7]       16
3      [3,5,7]              [3,  5,  7]       15
2      [2,3,5,7]          [2,3,  5,  7]       17
1      [1,2,3,5,7]      [1,2,3,  5,  7]       18
---------------------------------------------------
The total sum is a(5) = 1+2^2+3^2+4+5^2+6+7^2 = 1+4+9+4+25+6+49 = 18+17+15+16+12+13+7 = 98.

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

Partial sums of A210969. Row sums of triangle A210971.

Cf. A135010, A138121, A182703, A194446, A210437, A210966.

cp's OEIS Frontend

A278355 a(n) = sum of the perimeters of the Ferrers boards of the partitions of n. Also, sum of the perimeters of the diagrams of the regions of the set of partitions of n.

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A299474 a(n) = 4*p(n), where p(n) is the number of partitions of n.

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A299475 a(n) is the number of vertices in the diagram of partitions of n (see example).

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A179862 An unrestricted partition statistic: sum of A179864 over row n.

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A299773 a(n) is the index of the partition that contains the divisors of n in the list of colexicographically ordered partitions of the sum of the divisors of n.

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A273140 Number of parts in the corner of size n X n of the modular table of partitions described in Comments.

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A278602 Sum of the perimeters of all regions of the n-th section of a modular table of partitions.

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A210969 Sum of all region numbers of all parts of the last section of the set of partitions of n.

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