A186114 -id:A186114 - cp's OEIS frontend

A211026 Number of segments needed to draw (on the infinite square grid) a diagram of regions and partitions of n.

4, 6, 8, 12, 16, 24, 32, 46, 62, 86, 114, 156, 204, 272, 354, 464, 596, 772, 982, 1256, 1586, 2006, 2512, 3152, 3918, 4874, 6022, 7438, 9132, 11210, 13686, 16700, 20288, 24622, 29768, 35956, 43276, 52032, 62372, 74678, 89168, 106350

Offset: 1

Omar E. Pol, Oct 29 2012

nonn

On the infinite square grid the diagram of regions of the set of partitions of n is represented by a rectangle with base = n and height = A000041(n). The rectangle contains n shells. Each shell contains regions. Each row of a region is a part. Each part of size k contains k cells. The number of regions equals the number of partitions of n (see illustrations in the links section). For a minimalist version see A139582. For the definition of "region of n" see A206437.

Omar E. Pol, Illustration of initial terms of A211026 and of A066186
Omar E. Pol, Illustration of the seven regions of 5

Cf. A000041, A052810, A135010, A139582, A141285, A186412, A186114, A187219, A193870, A194446, A194447, A206437, A211009

a(n) = 2*A000041(n) + 2 = 2*A052810(n) = A139582(n) + 2.

a(18) corrected by Georg Fischer, Apr 11 2024

A228347 Triangle of regions and compositions of the positive integers (see Comments lines for definition).

Original entry on oeis.org

1, 1, 2, 0, 0, 1, 1, 1, 2, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Omar E. Pol, Aug 26 2013

Triangle read by rows in which row n lists A129760(n) zeros followed by the A006519(n) elements of the row A001511(n) of triangle A090996, n >= 1.

The equivalent sequence for partitions is A186114.

			----------------------------------------------------------
.             Diagram                Triangle
Compositions    of            of compositions (rows)
of 5          regions          and regions (columns)
----------------------------------------------------------
.            _ _ _ _ _
5           |_        |                                 5
1+4         |_|_      |                               1 4
2+3         |_  |     |                             2 0 3
1+1+3       |_|_|_    |                           1 1 0 3
3+2         |_    |   |                         3 0 0 0 2
1+2+2       |_|_  |   |                       1 2 0 0 0 2
2+1+2       |_  | |   |                     2 0 1 0 0 0 2
1+1+1+2     |_|_|_|_  |                   1 1 0 1 0 0 0 2
4+1         |_      | |                 4 0 0 0 0 0 0 0 1
1+3+1       |_|_    | |               1 3 0 0 0 0 0 0 0 1
2+2+1       |_  |   | |             2 0 2 0 0 0 0 0 0 0 1
1+1+2+1     |_|_|_  | |           1 1 0 2 0 0 0 0 0 0 0 1
3+1+1       |_    | | |         3 0 0 0 1 0 0 0 0 0 0 0 1
1+2+1+1     |_|_  | | |       1 2 0 0 0 1 0 0 0 0 0 0 0 1
2+1+1+1     |_  | | | |     2 0 1 0 0 0 1 0 0 0 0 0 0 0 1
1+1+1+1+1   |_|_|_|_|_|   1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1
.
For the positive integer k consider the first 2^(k-1) rows of triangle, as shown below. The positive terms of the n-th row are the parts of the n-th region of the diagram of regions of the set of compositions of k. The positive terms of the n-th column are the parts of the n-th composition of k, with compositions in colexicographic order.
Triangle begins:
1;
1,2;
0,0,1;
1,1,2,3;
0,0,0,0,1;
0,0,0,0,1,2;
0,0,0,0,0,0,1;
1,1,1,1,2,2,3,4;
0,0,0,0,0,0,0,0,1;
0,0,0,0,0,0,0,0,1,2;
0,0,0,0,0,0,0,0,0,0,1;
0,0,0,0,0,0,0,0,1,1,2,3;
0,0,0,0,0,0,0,0,0,0,0,0,1;
0,0,0,0,0,0,0,0,0,0,0,0,1,2;
0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
...

Mirror of A228348. Column 1 is A036987. Also column 1 gives A209229, n >= 1. Right border gives A001511. Positive terms give A228349.

Cf. A001792, A001787, A006519, A011782, A065120, A096996, A129760, A186114, A187816, A187818, A206437, A228350, A228351, A228366, A228367, A228370, A228371, A228525, A228526.

A182276 Sum of all parts minus the total number of parts of the shell model of partitions with n regions.

Original entry on oeis.org

0, 1, 3, 4, 8, 10, 15, 16, 20, 22, 31, 33, 38, 41, 51, 52, 56, 58, 67, 71, 74, 90, 92, 97, 100, 110, 112, 119, 123, 142, 143, 147, 149, 158, 162, 165, 181, 184, 192, 197, 201, 228, 230, 235, 238, 248, 250, 257, 261, 280, 284, 287, 299, 305, 310, 341

Offset: 1

Omar E. Pol, Apr 23 2012

nonn

For the definition of "region of n" see A206437.

			Written has a triangle:
0,
1,
3,
4,    8;
10,  15;
16,  20, 22, 31;
33,  38, 41, 51;
52,  56, 58, 67, 71, 74, 90;
92,  97,100,110,112,119,123,142;
143,147,149,158,162,165,181,184,192,197,201,228;
230,235,238,248,250,257,261,280,284,287,299,305,310,341;

Row j has length A187219(j). Right border gives A196087.

Cf. A000041, A066186, A006128, A135010, A138121, A182181, A182244, A186114, A206437, A207035.

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
reg = {}; l = {};
For[j = 1, j <= 56, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  t = Take[Reverse[First /@ lex[mx]], j - i];
  AppendTo[reg, Total@t - Length@t]
  ];
Accumulate@reg  (* Robert Price, Jul 25 2020 *)

a(n) = A182244(n) - A182181(n).

a(A000041(n)) = A196087(n).

A182377 Total sum of positive ranks of all regions in the last shell of n.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 8, 14, 21, 32, 45, 67, 91

Offset: 1

Omar E. Pol, Apr 29 2012

The rank of a region of n is the largest part minus the number of parts. For the definition of "region of n" see A206437. For the definition of "last shell of n" see A135010.

a(n) is also the sum of positive integers in row n of triangle A194447. First differs from A000094 at a(12).

			For n = 7 the last shell of 7 contains four regions: [3],[5,2],[4],[7,3,2,2,1,1,1,1,1,1,1,1,1,1,1] so we have:
----------------------------------------------------------
.        Largest    Number
Region     part    of parts    Rank
----------------------------------------------------------
.  1        3         1          2
.  2        5         2          3
.  3        4         1          3
.  4        7        15         -8
.
The sum of positive ranks is a(7) = 2 + 3 + 3 = 8.

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

Cf. A000094, A135010, A138121, A182699, A182709, A183152, A186114, A187219, A194436-A194439, A194447-A194448, A196025, A206437, A209616.

A183012 a(n) = 24*A138879(n) - A187219(n).

Original entry on oeis.org

23, 71, 119, 262, 358, 740, 932, 1697, 2248, 3588, 4690, 7371, 9312, 13814, 17959, 25289, 32406, 45056, 57015, 77383, 98043, 129678, 163451, 214120, 267217, 344786, 429842, 547308, 677897, 856601, 1054330, 1320077

Offset: 1

Omar E. Pol, Jan 22 2011

Partial sums give the positive terms of A183011, the numerators of the Bruinier-Ono formula for the partition function.

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

Cf. A000041, A002865, A135010, A138879, A183010, A183011, A186114, A206437.

A066186 := proc(n) n*combinat[numbpart](n) ; end proc:
A138879 := proc(n) A066186(n)-A066186(n-1) ; end proc:
A002865 := proc(n) if n = 0 then 1; else combinat[numbpart](n)-combinat[numbpart](n-1) ; end if; end proc:
A183012 := proc(n) if n = 1 then 23; else 24*A138879(n)-A002865(n) ; end if; end proc:
seq(A183012(n),n=1..80) ; # R. J. Mathar, Jan 27 2011

A211980 Triangle read by rows: T(n,k) = total number of regions in the last n-k+1 shells of n.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 5, 4, 3, 2, 7, 6, 5, 4, 2, 11, 10, 9, 8, 6, 4, 15, 14, 13, 12, 10, 8, 4, 22, 21, 20, 19, 17, 15, 11, 7, 30, 29, 28, 27, 25, 23, 19, 15, 8, 42, 41, 40, 39, 37, 35, 31, 27, 20, 12, 56, 55, 54, 53, 51, 49, 45, 41, 34, 26, 14, 77, 76, 75

Offset: 1

Omar E. Pol, Apr 27 2012

The set of partitions of n contains n shells and A000041(n) regions. For the definition of "last section of n" see A135010. For the definition of "region of n" see A206437.

			Triangle begins:
1;
2,   1;
3,   2,  1;
5,   4,  3,  2;
7,   6,  5,  4,  2;
11, 10,  9,  8,  6,  4;
15, 14, 13, 12, 10,  8,  4;
22, 21, 20, 19, 17, 15, 11,  7;
30, 29, 28, 27, 25, 23, 19, 15,  8;
42, 41, 40, 39, 37, 35, 31, 27, 20, 12;
56, 55, 54, 53, 51, 49, 45, 41, 34, 26, 14;
77, 76, 75, 74, 72, 70, 66, 62, 55, 47, 35, 21;

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

Mirror of triangle A211990. Column 1 is A000041, n >= 1. Right border is A187219.

Cf. A135010, A138121, A182703, A186114, A206437, A212000, A212001.

T(n,1) = A000041(n).

T(n,k) = A000041(n) - A000041(k-1), 1

T(n,k) = Sum_{j=k..n} A187219(j).

A225596 Sum of largest parts of all partitions of n plus n. Also, total number of parts in all partitions of n plus n.

Original entry on oeis.org

0, 2, 5, 9, 16, 25, 41, 61, 94, 137, 202, 286, 411, 569, 794, 1083, 1479, 1982, 2662, 3517, 4650, 6073, 7921, 10229, 13198, 16876, 21548, 27321, 34573, 43482, 54593, 68166, 84959, 105399, 130496, 160911, 198050, 242849, 297239, 362626, 441586, 536145

Offset: 0

Omar E. Pol, Aug 01 2013

nonn

a(n) is also the number of horizontal toothpicks (or the total length of all horizontal boundary segments) in the diagram of regions of the set of partitions of n, see example. A093694(n) is the number of vertical toothpicks in the diagram. See also A225610. For a minimalist version of the diagram see A211978. For the definition of "region" see A206437.

			For n = 7 the sum of largest parts of all partitions of 7 plus 7 is (7+4+5+3+6+3+4+2+5+3+4+2+3+2+1) + 7 = 54 + 7 = 61. On the other hand the number of toothpicks in horizontal direction in the diagram of regions of the set of partitions of 7 is equal to 61, so a(7) = 61.
.
.                  Diagram of regions       Horizontal
Partitions         and partitions of 7      toothpicks
of 7
.                     _ _ _ _ _ _ _
7                    |_ _ _ _      |             7
4+3                  |_ _ _ _|_    |             4
5+2                  |_ _ _    |   |             5
3+2+2                |_ _ _|_ _|_  |             3
6+1                  |_ _ _      | |             6
3+3+1                |_ _ _|_    | |             3
4+2+1                |_ _    |   | |             4
2+2+2+1              |_ _|_ _|_  | |             2
5+1+1                |_ _ _    | | |             5
3+2+1+1              |_ _ _|_  | | |             3
4+1+1+1              |_ _    | | | |             4
2+2+1+1+1            |_ _|_  | | | |             2
3+1+1+1+1            |_ _  | | | | |             3
2+1+1+1+1+1          |_  | | | | | |             2
1+1+1+1+1+1+1        |_|_|_|_|_|_|_|             1
.                                                7
.                                              _____
.                                       Total   61
.

Cf. A000041, A006128, A093694, A135010, A139250, A141285, A186114, A194446, A194447, A206437, A211978, A220517, A225600, A225610.

a(n) = A006128(n) + n = A225610(n) - A093694(n).

a(n) = n + sum_{k=1..A000041(n)} A141285(k), n >= 1.

A225598 Triangle read by rows: T(n,k) = sum of all parts of all regions of the set of partitions of n whose largest part is k.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 1, 5, 5, 9, 1, 5, 8, 9, 12, 1, 7, 11, 15, 12, 20, 1, 7, 14, 19, 19, 20, 25, 1, 9, 17, 29, 24, 33, 25, 38, 1, 9, 23, 33, 36, 42, 39, 38, 49, 1, 11, 26, 47, 46, 61, 49, 61, 49, 69, 1, 11, 32, 55, 63, 76, 70, 76, 76, 69, 87, 1, 13, 38, 73, 78, 110, 87, 111, 95, 108, 87, 123

Offset: 1

Omar E. Pol, Aug 02 2013

For the definition of region see A206437.

T(n,k) is also the sum of all 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 sum of all parts is 3 + 1 + 1 + 3 = 8, so T(5,3) = 8.
.
.    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     5       8         9         12
.
Triangle begins:
1;
1,  3;
1,  3,  5;
1,  5,  5,  9;
1,  5,  8,  9, 12;
1,  7, 11, 15, 12,  20;
1,  7, 14, 19, 19,  20, 25;
1,  9, 17, 29, 24,  33, 25,  38;
1,  9, 23, 33, 36,  42, 39,  38, 49;
1, 11, 26, 47, 46,  61, 49,  61, 49,  69;
1, 11, 32, 55, 63,  76, 70,  76, 76,  69, 87;
1, 13, 38, 73, 78, 110, 87, 111, 95, 108, 87, 123;

Column 1 is A000012. Column 2 are the numbers >= 3 of A109613. Row sums give A066186. Right border gives A046746. Second right border gives A046746.

Cf. A000041, A066186, A135010, A141285, A186114, A186412, A187219, A194446, A206437, A207779, A211978, A225597, A225600, A225610.

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

Omar E. Pol, Feb 10 2018

nonn

For n >= 1, a(n) is also the number of vertices in the minimalist diagram of partitions of n, in which A139582(n) is the number of line segments and A000041(n) is the number of open regions (see example).

			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.

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

k times partition numbers: A000041 (k=1), A139582 (k=2), this sequence (k=3), A299474 (k=4).

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

a(n) = 3*A000041(n) = A000041(n) + A139582(n).

a(n) = A299475(n) - 1, n >= 1.

A299774 Irregular triangle read by rows in which row n lists the indices of the partitions into equal parts in the list of colexicographically ordered partitions of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 5, 1, 7, 1, 8, 10, 11, 1, 15, 1, 16, 21, 22, 1, 27, 30, 1, 31, 41, 42, 1, 56, 1, 57, 69, 73, 76, 77, 1, 101, 1, 102, 134, 135, 1, 160, 172, 176, 1, 177, 221, 230, 231, 1, 297, 1, 298, 353, 380, 384, 385, 1, 490, 1, 491, 604, 615, 626, 627, 1

Offset: 1

Omar E. Pol, Mar 29 2018

Note that n is one of the partitions of n into equal parts.
If n is even then row n ending in [p(n) - 1, p(n)], where p(n) = A000041(n).
T(n,k) > p(n - 1), if 1 < k <= A000005(n).
Removing the 1's then all terms of the sequence are in increasing order.
If n is even then row n starts with [1, p(n - 1) + 1]. - David A. Corneth and Omar E. Pol, Aug 26 2018

			Triangle begins:
  1;
  1,   2;
  1,   3;
  1,   4,   5;
  1,   7;
  1,   8,  10,  11;
  1,  15;
  1,  16,  21,  22;
  1,  27,  30;
  1,  31,  41,  42;
  1,  56;
  1,  57,  69,  73,  76,  77;
  1, 101;
  1, 102, 134, 135;
  1, 160, 172, 176;
  ...
For n = 6 the partitions of 6 into equal parts are [1, 1, 1, 1, 1, 1], [2, 2, 2], [3, 3] and [6]. Then we have that in the list of colexicographically ordered partitions of 6 these partitions are in the rows 1, 8, 10 and 11 respectively as shown below, so the 6th row of the triangle is [1, 8, 10, 11].
-------------------------------------------------------------
   p      Diagram        Partitions of 6
-------------------------------------------------------------
        _ _ _ _ _ _
   1   |_| | | | | |    [1, 1, 1, 1, 1, 1]  <--- equal parts
   2   |_ _| | | | |    [2, 1, 1, 1, 1]
   3   |_ _ _| | | |    [3, 1, 1, 1]
   4   |_ _|   | | |    [2, 2, 1, 1]
   5   |_ _ _ _| | |    [4, 1, 1]
   6   |_ _ _|   | |    [3, 2, 1]
   7   |_ _ _ _ _| |    [5, 1]
   8   |_ _|   |   |    [2, 2, 2]  <--- equal parts
   9   |_ _ _ _|   |    [4, 2]
  10   |_ _ _|     |    [3, 3]  <--- equal parts
  11   |_ _ _ _ _ _|    [6]  <--- equal parts
.

Row n has length A000005(n).
Right border gives A000041, n >= 1.
Column 1 gives A000012.
Records give A317296.
Cf. A211992 (partitions in colexicographic order).
Cf. A027750, A135010, A141285, A186114, A186412, A193870, A194446, A194447, A211978, A206437, A299474, A299475, A299773, A299775.

row(n) = {if(n == 1, return([1])); my(nd = numdiv(n), res = vector(nd)); res[1] = 1; res[nd] = numbpart(n); if(nd > 2, t = nd - 1; p = vecsort(partitions(n)); forstep(i = #p - 1, 2, -1, if(p[i][1] == p[i][#p[i]], res[t] = i; t--; if(t==1, return(res)))), return(res))} \\ David A. Corneth, Aug 17 2018

Terms a(46) and beyond from David A. Corneth, Aug 16 2018

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cp's OEIS Frontend

A211026 Number of segments needed to draw (on the infinite square grid) a diagram of regions and partitions of n.

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A228347 Triangle of regions and compositions of the positive integers (see Comments lines for definition).

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A182276 Sum of all parts minus the total number of parts of the shell model of partitions with n regions.

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A182377 Total sum of positive ranks of all regions in the last shell of n.

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A183012 a(n) = 24*A138879(n) - A187219(n).

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Maple

A211980 Triangle read by rows: T(n,k) = total number of regions in the last n-k+1 shells of n.

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A225596 Sum of largest parts of all partitions of n plus n. Also, total number of parts in all partitions of n plus n.

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A225598 Triangle read by rows: T(n,k) = sum of all parts of all regions of the set of partitions of n whose largest part is k.

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A299473 a(n) = 3*p(n), where p(n) is the number of partitions of n.

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A299774 Irregular triangle read by rows in which row n lists the indices of the partitions into equal parts in the list of colexicographically ordered partitions of n.

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