A299474 -id:A299474 - cp's OEIS frontend

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

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).

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

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.

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

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

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

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