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.
Original entry on oeis.org
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
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.
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
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.
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
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
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.
Cf.
A000041,
A135010,
A139582,
A141285,
A182181,
A186114,
A193870,
A194446,
A194447,
A206437,
A207779,
A220482,
A220517,
A273140,
A278355,
A278602,
A299474.
A228349
Triangle read by rows: T(j,k) is the k-th part in nondecreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).
Original entry on oeis.org
1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 3, 4, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 3, 4, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1
Offset: 1
----------------------------------------------------------
. Diagram Triangle
Compositions of of compositions (rows)
of 5 regions and regions (columns)
----------------------------------------------------------
. _ _ _ _ _
5 |_ | 5
1+4 |_|_ | 1 4
2+3 |_ | | 2 3
1+1+3 |_|_|_ | 1 1 3
3+2 |_ | | 3 2
1+2+2 |_|_ | | 1 2 2
2+1+2 |_ | | | 2 1 2
1+1+1+2 |_|_|_|_ | 1 1 1 2
4+1 |_ | | 4 1
1+3+1 |_|_ | | 1 3 1
2+2+1 |_ | | | 2 2 1
1+1+2+1 |_|_|_ | | 1 1 2 1
3+1+1 |_ | | | 3 1 1
1+2+1+1 |_|_ | | | 1 2 1 1
2+1+1+1 |_ | | | | 2 1 1 1
1+1+1+1+1 |_|_|_|_|_| 1 1 1 1 1
.
Written as an irregular triangle in which row n lists the parts of the n-th region the sequence begins:
1;
1,2;
1;
1,1,2,3;
1;
1,2;
1;
1,1,1,1,2,2,3,4;
1;
1,2;
1;
1,1,2,3;
1;
1,2;
1;
1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
...
Alternative interpretation of this sequence:
Triangle read by rows in which row r lists the regions of the last section of the set of compositions of r:
[1];
[1,2];
[1],[1,1,2,3];
[1],[1,2],[1],[1,1,1,1,2,2,3,4];
[1],[1,2],[1],[1,1,2,3],[1],[1,2],[1],[1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5];
Cf.
A001787,
A001792,
A011782,
A029837,
A045623,
A065120,
A070939,
A090996,
A186114,
A187816,
A187818,
A206437,
A220482,
A228347,
A228348,
A228350,
A228351,
A228366,
A228367,
A228370,
A228371,
A228525,
A228526.
-
Table[Map[Length@ TakeWhile[IntegerDigits[#, 2], # == 1 &] &, Range[2^(# - 1), 2^# - 1]] &@ IntegerExponent[2 n, 2], {n, 32}] // Flatten (* Michael De Vlieger, May 23 2017 *)
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
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
.
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
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.
A299473
a(n) = 3*p(n), where p(n) is the number of partitions of n.
Original entry on oeis.org
3, 3, 6, 9, 15, 21, 33, 45, 66, 90, 126, 168, 231, 303, 405, 528, 693, 891, 1155, 1470, 1881, 2376, 3006, 3765, 4725, 5874, 7308, 9030, 11154, 13695, 16812, 20526, 25047, 30429, 36930, 44649, 53931, 64911, 78045, 93555, 112014, 133749, 159522, 189783, 225525, 267402, 316674, 374262, 441819, 520575, 612678
Offset: 0
Construction of a minimalist version 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)
A000041(n): 1 2 3 5 7 11 (open regions)
A139582(n): 2 4 6 10 14 22 (line segments)
a(n)......: 3 6 9 15 21 33 (vertices)
-----------------------------------------------------------------------------------
r p(n)
-----------------------------------------------------------------------------------
.
1 .... 1 .... _| _| | _| | | _| | | | _| | | | | _| | | | | |
2 .... 2 ......... _ _| _ _| | _ _| | | _ _| | | | _ _| | | | |
3 .... 3 ................ _ _ _| _ _ _| | _ _ _| | | _ _ _| | | |
4 _ _| | _ _| | | _ _| | | |
5 .... 5 ......................... _ _ _ _| _ _ _ _| | _ _ _ _| | |
6 _ _ _| | _ _ _| | |
7 .... 7 .................................... _ _ _ _ _| _ _ _ _ _| |
8 _ _| | |
9 _ _ _ _| |
10 _ _ _| |
11 .. 11 ................................................. _ _ _ _ _ _|
.
The r-th horizontal line segment has length A141285(r).
The r-th vertical line segment has length A194446(r).
An infinite diagram is a minimalist table of all partitions of all positive integers.
Cf.
A135010,
A141285,
A182181,
A186114,
A193870,
A194446,
A194447,
A206437,
A207779,
A220482,
A220517,
A273140,
A278355,
A278602,
A299475.
Showing 1-7 of 7 results.
Comments