A193870 -id:A193870 - cp's OEIS frontend

A194439 Number of regions in the set of partitions of n that contain only one part.

1, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297

Offset: 1

Omar E. Pol, Nov 28 2011

It appears that this is 1 together with A000041. - Omar E. Pol, Nov 29 2011

For the definition of "region" see A206437. See also A186114 and A193870.

			For n = 5 the seven regions of 5 in nondecreasing order are the sets of positive integers of the rows as shown below:
   1;
   1, 2;
   1, 1, 3;
   0, 0, 0, 2;
   1, 1, 1, 2, 4;
   0, 0, 0, 0, 0, 3;
   1, 1, 1, 1, 1, 2, 5;
   ...
There are three regions that contain only one positive part, so a(5) = 3.
Note that in every column of the triangle the positive integers are also the parts of one of the partitions of 5.

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

Column 1 of A194438.

Cf. A000041, A002865, A027336, A135010, A138121, A186114, A186412, A193870, A194436, A194437, A194446, A194447, A206437.

It appears that a(n) = A000041(n-2), if n >= 2. - Omar E. Pol, Nov 29 2011

It appears that a(n) = A000041(n) - A027336(n), if n >= 2. - Omar E. Pol, Nov 30 2011

Definition clarified by Omar E. Pol, May 21 2021

A183152 Irregular triangle read by rows in which row n lists the emergent parts of all partitions of n, or 0 if such parts do not exist.

Original entry on oeis.org

0, 0, 0, 0, 2, 3, 2, 4, 2, 3, 3, 5, 2, 4, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 3, 5, 2, 4, 7, 3, 2, 2, 3, 6, 3, 5, 2, 4, 2, 3, 6, 3, 2, 2, 5, 4, 8, 4, 3, 2, 2, 2, 2, 4, 7, 3, 6, 5, 3, 5, 2, 4, 7, 3, 2, 2, 3, 6, 3, 5, 9, 4, 3, 3, 2, 2, 2, 2, 5, 4, 8, 4, 3, 7, 6

Offset: 0

Omar E. Pol, Aug 07 2011

For the definition of "emergent part" see A182699 and also A182709.

Also [0, 0, 0, 0] followed by the positive integers of the rows that contain zeros in the triangle A193870. For another version see A193827. - Omar E. Pol, Aug 12 2011

			If written as a triangle:
0,
0,
0,
0,
2,
3,
2,4,2,3,
3,5,2,4,
2,4,2,3,6,3,2,2,5,4,
3,5,2,4,7,3,2,2,3,6,3,5,
2,4,2,3,6,3,2,2,5,4,8,4,3,2,2,2,2,4,7,3,6,5,
3,5,2,4,7,3,2,2,3,6,3,5,9,4,3,3,2,2,2,2,5,4,8,4,3,7,6

Row n has length A182699(n). Row sums give A182709.

Cf. A135010, A138121, A182707.

A167392 Characteristic function of partition numbers.

Original entry on oeis.org

0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Nov 03 2009

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for characteristic functions

Cf. A167393.

For n >= 1, column 1 of A186114, also right border of A193870. - Omar E. Pol, Aug 14 2011

import Data.List.Ordered (member)
a167392 = fromEnum . flip member a000041_list
-- Reinhard Zumkeller, Nov 03 2015

nmax = 14;
(* nmax=14 gives P(nmax)+1 = 136 terms; nmax=33 gives 10144 terms *)
PP = Table[PartitionsP[n], {n, 0, nmax}];
a[n_] := Boole[MemberQ[PP, n]];
Table[a[n], {n, 0, PartitionsP[nmax]}] (* Jean-François Alcover, Mar 02 2019 *)

a(n) = {k=0; while ((pk=numbpart(k)) != n, if (pk > n, return(0)); k++); return (1);} \\ Michel Marcus, Nov 03 2015

a(A000041(n)) = 1; a(A167376(n)) = 0.

A211009 Triangle read by rows: T(n,k) = number of cells in the k-column of the n-th region of j in the list of colexicographically ordered partitions of j, if 1<=n<=A000041(j), 1<=k<=A141285(n).

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 1, 2, 7, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 4, 11, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 15, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 7, 22

Offset: 1

Omar E. Pol, Oct 21 2012

Also the finite sequence a(1)..a(r), where a(r) is a record in the sequence, is also a finite triangle read by rows: T(n,k) = number of cells in the k-column of the n-th region of the integer whose number of partitions is equal to a(r).
T(n,k) is also 1 plus the number of holes between T(n,k) and the previous member in the column k of triangle.
T(n,k) is also the height of the column mentioned in the definition, in a three-dimensional model of the set of partitions of j, in which the regions appear rotated 90 degrees and where the pivots are the largest part of every region (see A141285). For the definition of "region" see A206437. - Omar E. Pol, Feb 06 2014

			The irregular triangle begins:
1;
1, 2;
1, 1, 3;
1, 1;
1, 1, 2, 5;
1, 1, 1;
1, 1, 1, 2, 7;
1, 1;
1, 1, 2, 2;
1, 1, 1;
1, 1, 1, 2, 4, 11;
1, 1, 1;
1, 1, 1, 2, 2;
1, 1, 1, 1;
1, 1, 1, 1, 2, 4, 15;
1, 1;
1, 1, 2, 2;
1, 1, 1;
1, 1, 1, 2, 4, 4;
1, 1, 1, 1, 1;
1, 1, 1, 1;
1, 1, 1, 1, 2, 3, 7, 22;
...
From _Omar E. Pol_, Feb 06 2014: (Start)
Illustration of initial terms:
.    _
.   |_|
.    1
.      _
.    _|_|
.   |_ _|
.    1 2
.        _
.       |_|
.    _ _|_|
.   |_ _ _|
.    1 1 3
.    _ _
.   |_ _|
.    1 1
.          _
.         |_|
.         |_|
.        _|_|
.    _ _|_ _|
.   |_ _ _ _|
.    1 1 2 5
.
(End)

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

Records give positive terms of A000041. Row n has length A141285(n). Row sums give A186412.

Cf. A040051, A135010, A182244, A182703, A186114, A187219, A193870, A194446, A206437.

Better definition from Omar E. Pol, Feb 06 2014

A194438 Triangle read by rows: T(n,k) is the number of regions of the set of partitions of n into k parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 3, 1, 1, 0, 1, 0, 1, 5, 2, 1, 0, 1, 0, 1, 0, 0, 0, 1, 7, 3, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 11, 4, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 15, 6, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0

Offset: 1

Omar E. Pol, Nov 28 2011

For the definition of "region" see A206437. See also A186114 and A193870. - Omar E. Pol, May 21 2021

			Triangle begins:
   1;
   1,1;
   1,1,1;
   2,1,1,0,1;
   3,1,1,0,1,0,1;
   5,2,1,0,1,0,1,0,0,0,1;
   7,3,1,0,1,0,1,0,0,0,1,0,0,0,1;
  11,4,1,1,1,0,1,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1;
...

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

Column 1 is A194439.
Row n has length A000041(n).
Row sums give A000041, n >= 1.
Cf. A008284, A135010, A138121, A186114, A186412, A193870, A194436, A194437, A194439, A194446, A194447, A206437.

Definition clarified by Omar E. Pol, May 21 2021

A194437 Triangle read by rows: T(n,k) = sum of parts in the k-th region of n.

Original entry on oeis.org

1, 1, 3, 1, 3, 5, 1, 3, 5, 2, 9, 1, 3, 5, 2, 9, 3, 12, 1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 3, 7, 4, 25, 1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 3, 7, 4, 25, 2, 6, 3, 13, 5, 4, 38, 1, 3, 5, 2, 9, 3, 12, 2, 6, 3, 20, 3, 7

Offset: 1

Omar E. Pol, Nov 27 2011

			Triangle begins:
1;
1,3;
1,3,5;
1,3,5,2,9;
1,3,5,2,9,3,12;
1,3,5,2,9,3,12,2,6,3,20;
1,3,5,2,9,3,12,2,6,3,20,3,7,4,25;
1,3,5,2,9,3,12,2,6,3,20,3,7,4,25,2,6,3,13,5,4,38;
...
Row n has length A000041(n). Row sums give A066186. Right border gives A046746. Records in every row give A046746. Rows converge to A186412.

Cf. A046746, A066186, A135010, A138121, A186114, A186412, A193870, A194436, A194438, A194439, A194446, A194447.

A220482 Triangle read by rows: T(j,k) in which row j lists the parts in nondecreasing order of the j-th region of the set of partitions of n, with 1<=j<=A000041(n).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jan 27 2013

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

			First 15 rows of the irregular triangle are
1;
1, 2;
1, 1, 3;
2;
1, 1, 1, 2, 4;
3;
1, 1, 1, 1, 1, 2, 5;
2;
2, 4;
3;
1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 6;
3;
2, 5;
4;
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 7;

Positive terms of A186114. Mirror of A206437.
Row j has length A194446(j). Row sums give A186412.
Cf. A135010, A141285, A193870.

A228350 Triangle read by rows: T(j,k) is the k-th part in nonincreasing 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, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 6, 5, 4, 4, 3, 3

Offset: 1

Omar E. Pol, Aug 20 2013

Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A065120, n >= 1.

The equivalent sequence for integer partitions is A206437.

			---------------------------------------------------------
.              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
.
Also the structure could be represented by an isosceles triangle in which the n-th diagonal gives the n-th region. For the composition of 4 see below:
.             _ _ _ _
.         4  |_      |                  4
.       1+3  |_|_    |                1   3
.       2+2  |_  |   |              2       2
.     1+1+2  |_|_|_  |            1   1       2
.       3+1  |_    | |          3               1
.     1+2+1  |_|_  | |        1   2               1
.     2+1+1  |_  | | |      2       1               1
.   1+1+1+1  |_|_|_|_|    1   1       1               1
.
Illustration of the four sections of the set of compositions of 4:
.                                      _ _ _ _
.                                     |_      |     4
.                                     |_|_    |   1+3
.                                     |_  |   |   2+2
.                       _ _ _         |_|_|_  | 1+1+2
.                      |_    |   3          | |     1
.             _ _      |_|_  | 1+2          | |     1
.     _      |_  | 2       | |   1          | |     1
.    |_| 1     |_| 1       |_|   1          |_|     1
.
.
Illustration of initial terms. The parts of the eight regions of the set of compositions of 4:
--------------------------------------------------------
\j:  1      2    3        4     5      6    7          8
k
--------------------------------------------------------
.  _    _ _    _    _ _ _     _    _ _    _    _ _ _ _
1 |_|1 |_  |2 |_|1 |_    |3  |_|1 |_  |2 |_|1 |_      |4
2        |_|1        |_  |2         |_|1        |_    |3
3                      | |1                       |   |2
4                      |_|1                       |_  |2
5                                                   | |1
6                                                   | |1
7                                                   | |1
8                                                   |_|1
.
Triangle begins:
1;
2,1;
1;
3,2,1,1;
1;
2,1;
1;
4,3,2,2,1,1,1,1;
1;
2,1;
1;
3,2,1,1;
1;
2,1;
1;
5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1;
...
.
Also triangle read by rows T(n,m) in which row n lists the parts of the n-th section of the set of compositions of the integers >= n, ordered by regions. Row lengths give A045623. Row sums give A001792 (see below):
[1];
[2,1];
[1],[3,2,1,1];
[1],[2,1],[1],[4,3,2,2,1,1,1,1];
[1],[2,1],[1],[3,2,1,1],[1],[2,1],[1],[5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1];

Main triangle: column 1 is A001511. Row j has length A006519(j). Row sums give A038712.

Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A135010, A141285, A187816, A187818, A193870, A206437, A228347, A228348, A228349, A228351, A228366, A228367, A228370, A228371, A228525, A228526.

T(j,k) = A065120(A001511(j)),k) = A001511(j) - A029837(k), 1<=k<=A006519(j), j>=1.

A194448 Number of parts > 1 in the n-th region of the shell model of partitions.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 3, 1, 1, 14, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 1, 21, 1, 2, 1, 4, 1, 2

Offset: 1

Omar E. Pol, Dec 10 2011

Also triangle read by rows: T(n,k) = number of parts > 1 in the k-th region of the last section of the set of partitions of n.

			Written as a triangle:
0;
1;
1;
1,2;
1,2;
1,2,1,4;
1,2,1,4;
1,2,1,4,1,1,7;
1,2,1,4,1,2,1,8;
1,2,1,4,1,1,7,1,2,1,1,12;
1,2,1,4,1,2,1,8,1,1,3,1,1,14;
1,2,1,4,1,1,7,1,2,1,1,12,1,2,1,4,1,2,1,1,21;

Row sums give A138135. Row n has length A187219(n).

Cf. A000041, A002865, A135010, A138121, A138137, A138879, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194446, A194447, A194449.

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

cp's OEIS Frontend

A194439 Number of regions in the set of partitions of n that contain only one part.

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A183152 Irregular triangle read by rows in which row n lists the emergent parts of all partitions of n, or 0 if such parts do not exist.

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A167392 Characteristic function of partition numbers.

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A211009 Triangle read by rows: T(n,k) = number of cells in the k-column of the n-th region of j in the list of colexicographically ordered partitions of j, if 1<=n<=A000041(j), 1<=k<=A141285(n).

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A194438 Triangle read by rows: T(n,k) is the number of regions of the set of partitions of n into k parts.

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A194437 Triangle read by rows: T(n,k) = sum of parts in the k-th region of n.

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A220482 Triangle read by rows: T(j,k) in which row j lists the parts in nondecreasing order of the j-th region of the set of partitions of n, with 1<=j<=A000041(n).

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A228350 Triangle read by rows: T(j,k) is the k-th part in nonincreasing 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).

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A194448 Number of parts > 1 in the n-th region of the shell model of partitions.

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

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