A138121 -id:A138121 - cp's OEIS frontend

A211994 A list of ordered partitions of the positive integers.

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

Offset: 1

Omar E. Pol, Aug 18 2012

The order of the partitions of the odd integers is the same as A026792. The order of the partitions of the even integers is the same as A211992.

			A table of partitions.
--------------------------------------------
.              Expanded       Geometric
Partitions     arrangement    model
--------------------------------------------
1;             1;             |*|
--------------------------------------------
1,1;           1,1;           |o|*|
2;             . 2;           |* *|
--------------------------------------------
3;             . . 3;         |* * *|
2,1;           . 2,1;         |o o|*|
1,1,1;         1,1,1;         |o|o|*|
--------------------------------------------
1,1,1,1;       1,1,1,1;       |o|o|o|*|
2,1,1;         . 2,1,1;       |o o|o|*|
3,1;           . . 3,1;       |o o o|*|
2,2;           . 2,. 2;       |* *|* *|
4;             . . . 4;       |* * * *|
--------------------------------------------
5;             . . . . 5;     |* * * * *|
3,2;           . . 3,. 2;     |* * *|* *|
4,1;           . . . 4,1;     |o o o o|*|
2,2,1;         . 2,. 2,1;     |o o|o o|*|
3,1,1;         . . 3,1,1;     |o o o|o|*|
2,1,1,1;       . 2,1,1,1;     |o o|o|o|*|
1,1,1,1,1;     1,1,1,1,1;     |o|o|o|o|*|
--------------------------------------------
1,1,1,1,1,1;   1,1,1,1,1,1;   |o|o|o|o|o|*|
2,1,1,1,1;     . 2,1,1,1,1;   |o o|o|o|o|*|
3,1,1,1;       . . 3,1,1,1;   |o o o|o|o|*|
2,2,1,1;       . 2,. 2,1,1;   |o o|o o|o|*|
4,1,1;         . . . 4,1,1;   |o o o o|o|*|
3,2,1;         . . 3,. 2,1;   |o o o|o o|*|
5,1;           . . . . 5,1;   |o o o o o|*|
2,2,2;         . 2,. 2,. 2;   |* *|* *|* *|
4,2;           . . . 4,. 2;   |* * * *|* *|
3,3;           . . 3,. . 3;   |* * *|* * *|
6;             . . . . . 6;   |* * * * * *|
--------------------------------------------
Note that * is a unitary element of every part of the last section of j.

Rows sums give A036042, n>=1.
Other versions are A026792, A211992, A211993. See also A211983, A211984, A211989, A211999. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A340031 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the j-th row of triangle A127093, where j = n - m + 1 and 1 <= m <= n.

Original entry on oeis.org

1, 1, 2, 1, 1, 0, 3, 1, 2, 1, 1, 1, 2, 0, 4, 1, 0, 3, 1, 2, 1, 2, 1, 1, 1, 1, 0, 0, 0, 5, 1, 2, 0, 4, 1, 0, 3, 1, 0, 3, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 0, 0, 6, 1, 0, 0, 0, 5, 1, 2, 0, 4, 1, 2, 0, 4, 1, 0, 3, 1, 0, 3, 1, 0, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Omar E. Pol, Dec 26 2020

Another version of A338156 which is the main sequence with further information about the correspondence divisor/part.

			Triangle begins:
[1];
[1,2],      [1];
[1,0,3],    [1,2],    [1],    [1];
[1,2,0,4],  [1,0,3],  [1,2],  [1,2],  [1],  [1],  [1];
[1,0,0,0,5],[1,2,0,4],[1,0,3],[1,0,3],[1,2],[1,2],[1,2],[1],[1],[1],[1],[1];
[...
Written as an irregular tetrahedron the first five slices are:
[1],
-------
[1, 2],
[1],
----------
[1, 0, 3],
[1, 2],
[1],
[1];
-------------
[1, 2, 0, 4],
[1, 0, 3],
[1, 2],
[1, 2],
[1],
[1],
[1];
----------------
[1, 0, 0, 0, 5],
[1, 2, 0, 4],
[1, 0, 3],
[1, 0, 3],
[1, 2],
[1, 2],
[1, 2],
[1],
[1],
[1],
[1],
[1];
.
The following table formed by three zones shows the correspondence between divisors and parts (n = 1..5):
.
|---|---------|-----|-------|---------|-----------|-------------|
| n |         |  1  |   2   |    3    |     4     |      5      |
|---|---------|-----|-------|---------|-----------|-------------|
| P |         |     |       |         |           |             |
| A |         |     |       |         |           |             |
| R |         |     |       |         |           |             |
| T |         |     |       |         |           |  5          |
| I |         |     |       |         |           |  3 2        |
| T |         |     |       |         |  4        |  4 1        |
| I |         |     |       |         |  2 2      |  2 2 1      |
| O |         |     |       |  3      |  3 1      |  3 1 1      |
| N |         |     |  2    |  2 1    |  2 1 1    |  2 1 1 1    |
| S |         |  1  |  1 1  |  1 1 1  |  1 1 1 1  |  1 1 1 1 1  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A181187 |  1  |  3 1  |  6 2 1  | 12 5 2 1  | 20 8 4 2 1  |
| L |         |  |  |  |/|  |  |/|/|  |  |/|/|/|  |  |/|/|/|/|  |
| I | A066633 |  1  |  2 1  |  4 1 1  |  7 3 1 1  | 12 4 2 1 1  |
| N |         |  *  |  * *  |  * * *  |  * * * *  |  * * * * *  |
| K | A002260 |  1  |  1 2  |  1 2 3  |  1 2 3 4  |  1 2 3 4 5  |
|   |         |  =  |  = =  |  = = =  |  = = = =  |  = = = = =  |
|   | A138785 |  1  |  2 2  |  4 2 3  |  7 6 3 4  | 12 8 6 4 5  |
|---|---------|-----|-------|---------|-----------|-------------|
.
|---|---------|-----|-------|---------|-----------|-------------|
|   | A127093 |  1  |  1 2  |  1 0 3  |  1 2 0 4  |  1 0 0 0 5  |
|   |---------|-----|-------|---------|-----------|-------------|
|   | A127093 |     |  1    |  1 2    |  1 0 3    |  1 2 0 4    |
|   |---------|-----|-------|---------|-----------|-------------|
| D | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| I | A127093 |     |       |  1      |  1 2      |  1 0 3      |
| V |---------|-----|-------|---------|-----------|-------------|
| I | A127093 |     |       |         |  1        |  1 2        |
| S | A127093 |     |       |         |  1        |  1 2        |
| O | A127093 |     |       |         |  1        |  1 2        |
| R |---------|-----|-------|---------|-----------|-------------|
| S | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|   | A127093 |     |       |         |           |  1          |
|---|---------|-----|-------|---------|-----------|-------------|
.
The table is essentially the same table of A338156 but here, in the lower zone, every row is A127093 instead of A027750.
.

Paolo Xausa, Table of n, a(n) for n = 1..11552 (rows 1..17 of the triangle, flattened)

Row sums give A066186.
Nonzero terms gives A338156.
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A066633, A127093, A135010, A138121, A138785, A176206, A181187, A182703, A207031, A207383, A211992, A221529, A221530, A221531, A221649, A221650, A237593, A245095, A302246, A302247, A336811, A337209,  A339106, A339258, A339278, A339304, A340011, A340032, A340035, A340061.

A127093row[n_]:=Table[Boole[Divisible[n,k]]k,{k,n}];
A340031row[n_]:=Flatten[Table[ConstantArray[A127093row[n-m+1],PartitionsP[m-1]],{m,n}]];
Array[A340031row,7] (* Paolo Xausa, Sep 28 2023 *)

A100818 For a given unrestricted partition pi, let P(pi)=lambda(pi), if mu(pi)=0. If mu(pi)>0 then let P(pi)=nu(pi), where nu(pi) is the number of parts of pi greater than mu(pi), mu(pi) is the number of ones in pi and lambda(pi) is the largest part of pi.

Original entry on oeis.org

1, 2, 1, 4, 3, 8, 7, 15, 15, 27, 29, 48, 53, 82, 94, 137, 160, 225, 265, 362, 430, 572, 683, 892, 1066, 1370, 1640, 2078, 2487, 3117, 3725, 4624, 5519, 6791, 8092, 9885, 11752, 14263, 16922, 20416, 24167, 29007, 34254, 40921, 48213, 57345, 67409, 79864

Offset: 1

David S. Newman, Jan 13 2005

nonn

Note that this is very similar to the "crank" of Andrews and Garvan. The number of partitions pi with P(pi) odd is the given sequence.
The sequence is the same as A087787 except for the value of a(1) (this was established by George Andrews, Jan 18 2005). If "even" is replace by "odd" in the definition of the sequence, the new sequence is almost identical except for two values and a shift to the right.
Also, positive integers of A182712. a(n) is also the number of 2's in the n-th row that contain a 2 as a part in the triangle of A138121 (note that rows 1 and 3 do not contain a 2 as a part). - Omar E. Pol, Nov 28 2010

			a(3)=1 because P(3)=3, P(2 1)=1 and P(1 1 1)=0.

G. C. Greubel, Table of n, a(n) for n = 1..1000
G. E. Andrews and F. Garvan, Dyson's crank of a partition, Bull. Amer. Math. Soc., 18 (1988), 167-171.

Cf. A000041, A087787, A135010, A138121, A182712.

Rest[ CoefficientList[ Series[x + 1/(1 + x) Product[1/(1 - x^n), {n, 50}], {x, 0, 50}], x]] (* Robert G. Wilson v, Feb 11 2005 *)

G.f.: x+(1/(1+x))* Product_{n>=1}(1/(1-x^n)). [corrected by Vaclav Kotesovec, Aug 29 2019]
a(n) = A000041(n) - a(n-1), for n>2. - Jon Maiga, Aug 29 2019 [corrected by Vaclav Kotesovec, Aug 29 2019]
a(n) = a(n-2) + A000041(n-1) - A000041(n-2), for n>=3. - Vaclav Kotesovec, Aug 29 2019

More terms from Robert G. Wilson v, Feb 11 2005

A138151 Irregular triangle read by rows in which rows 1..n (when read together) list all the parts in the partitions of n and row n starts with the partitions of n that do not contain 1 as a part (in the order used for A080577).

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 4, 2, 2, 1, 1, 1, 5, 3, 2, 1, 1, 1, 1, 1, 6, 4, 2, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 7, 5, 2, 4, 3, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 6, 2, 5, 3, 4, 4, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 7, 2, 6, 3, 5, 4, 5, 2, 2, 4, 3, 2, 3, 3, 3, 3, 2, 2

Offset: 1

Omar E. Pol, Mar 21 2008

The remainder of row n is necessarily A000041(n-1) 1's.
Previous name: A shell model of partitions. Row n lists the parts of the last section of the set of partitions of n.
Row n lists the nonzero terms of the n-th row of A138136 together with A000041(n-1) 1's.
Row n is also the n-th row of A138138 in reverse order.

			Triangle begins:
1
2,1
3,1,1
4,2,2,1,1,1
5,3,2,1,1,1,1,1,
6,4,2,3,3,2,2,2,1,1,1,1,1,1,1
7,5,2,4,3,3,2,2,1,1,1,1,1,1,1,1,1,1,1
8,6,2,5,3,4,4,4,2,2,3,3,2,2,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
9,7,2,6,3,5,4,5,2,2,4,3,2,3,3,3,3,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

Robert Price, Table of n, a(n) for n = 1..4630, 20 rows.
Omar E. Pol, Illustration of the rows 1..6 of the triangle

Mirror of A138138.
Row lengths give A138137.
Row sums give A138879.
Column 1 gives A000027.
Right border gives A000012.
Another version is A138121 which is the mirror of A135010.
Cf. A000041, A080577, A138136, A139094, A139100.

Table[Cases[IntegerPartitions[n], x_ /; Last[x] != 1] ~Join~ConstantArray[{1}, PartitionsP[n - 1]], {n, 8}] // Flatten (* Robert Price, May 22 2020 *)

New name and comments edited by Peter Munn and Omar E. Pol, Jul 25 2025

A139100 Triangle read by rows: row n lists all partitions of n in the order produced by the shell model of partitions A138151.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 15 2008

See the integrated diagram of partitions in the entry A138138.
See A138151 for more information.
First 43 members = A026792.

			Triangle begins:
{(1)}
{(2), (1, 1)}
{(3), (2, 1), (1, 1, 1)}
{(4), (2, 2), (3, 1), (2, 1, 1), (1, 1, 1, 1)}
{(5), (3, 2), (4, 1), (2, 2, 1), (3, 1, 1), (2, 1, 1, 1), (1, 1, 1, 1, 1)}

Robert Price, Table of n, a(n) for n = 1..7043, 17 rows.

Cf. A000041, A006128, A026792, A036036, A080576, A080577, A135010, A138121, A138135, A138136, A138137, A138138, A138151.

Table[If[n == 1, ConstantArray[{1}, i - n + 1],
   Map[(Join[#, ConstantArray[{1}, i - n]]) &,
    Cases[IntegerPartitions[n], x_ /; Last[x] != 1]]], {i, 7}, {n, i, 1, -1}]  // Flatten(* Robert Price, May 28 2020 *)

A168018 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 5, 2, 1, 7, 1, 11, 3, 2, 1, 15, 1, 22, 5, 2, 1, 30, 3, 1, 42, 7, 2, 1, 56, 1, 77, 11, 5, 3, 2, 1, 101, 1, 135, 15, 2, 1, 176, 7, 3, 1, 231, 22, 5, 2, 1, 297, 1, 385, 30, 11, 3, 2, 1, 490, 1, 627, 42, 7, 5, 2, 1, 792, 15, 3, 1, 1002, 56, 2, 1, 1255, 1, 1575, 77, 22, 11, 5, 3, 2

Offset: 1

Omar E. Pol, Nov 22 2009

Positive values of triangle A168021.
Note that column 1 lists the numbers of partitions A000041(n).
Row n has A000005(n) terms.
Also, it appears that row n lists the partition numbers of the divisors of n, in decreasing order. [Omar E. Pol, Nov 23 2009]

			For example:
Consider row 8: (22, 5, 2, 1). The divisors of 8 are 1, 2, 4, 8 (see A027750). Also, there are 22 partitions of 8 into parts divisible by 1 (A000041(8)=22); 5 partitions of 8 into parts divisible by 2: {(8),(6+2),(4+4),(4+2+2),(2+2+2+2)}; 2 partitions of 8 into parts divisible by 4: {(8),(4+4)}; and 1 partition of 8 into parts divisible by 8. Then row 8 is formed by 22, 5, 2, 1.
Triangle begins:
1;
2, 1;
3, 1;
5, 2, 1;
7, 1;
11, 3, 2, 1;
15, 1;
22, 5, 2, 1;
30, 3, 1;
42, 7, 2, 1;
56, 1;
77, 11, 5, 3, 2, 1;

Omar E. Pol, Illustration of the partitions of n, for n = 1 .. 9

Row sums give A047968.

Cf. A000005, A000041, A027750, A135010, A138121, A168016, A168017, A168019, A168020, A168021.

A168018 := proc(n) local dvs,p,i,d,a,pp,divs,par; dvs := sort(convert(numtheory[divisors](n),list)) ; p := combinat[partition](n) ; for i from 1 to nops(dvs) do d := op(i,dvs) ; a := 0 ; for pp in p do divs := true; for par in pp do if par mod d <> 0 then divs := false; end if; end do ; if divs then a := a+1 ; end if; end do ; printf("%d,",a) ; end do ; end proc: for n from 1 to 40 do A168018(n) ; end do : # R. J. Mathar, Feb 05 2010

Terms beyond row 12 from R. J. Mathar, Feb 05 2010

A182713 Number of 3's in the last section of the set of partitions of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 2, 3, 6, 6, 10, 14, 18, 24, 35, 42, 58, 76, 97, 124, 164, 202, 261, 329, 412, 514, 649, 795, 992, 1223, 1503, 1839, 2262, 2741, 3346, 4056, 4908, 5919, 7150, 8568, 10297, 12320, 14721, 17542, 20911, 24808, 29456, 34870, 41232, 48652, 57389

Offset: 1

Omar E. Pol, Nov 28 2010

nonn

Also number of 3's in all partitions of n that do not contain 1 as a part.
Also 0 together with the first differences of A024787. - Omar E. Pol, Nov 13 2011
a(n) is the number of partitions of n having fewer 1s than 2s; e.g., a(7) counts these 3 partitions: [5, 2], [3, 2, 2], [2, 2, 2, 1]. - Clark Kimberling, Mar 31 2014
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

			a(7) = 2 counts the 3's in 7 = 4+3 = 3+2+2. The 3's in 7 = 3+3+1 = 3+2+1+1 = 3+1+1+1+1 do not count.
From _Omar E. Pol_, Oct 27 2012: (Start)
--------------------------------------
Last section                   Number
of the set of                    of
partitions of 7                 3's
--------------------------------------
7 .............................. 0
4 + 3 .......................... 1
5 + 2 .......................... 0
3 + 2 + 2 ...................... 1
.   1 .......................... 0
.       1 ...................... 0
.       1 ...................... 0
.           1 .................. 0
.       1 ...................... 0
.           1 .................. 0
.           1 .................. 0
.               1 .............. 0
.               1 .............. 0
.                   1 .......... 0
.                       1 ...... 0
------------------------------------
.       5 - 3 =                  2
.
In the last section of the set of partitions of 7 the difference between the sum of the third column and the sum of the fourth column is 5 - 3 = 2 equaling the number of 3's, so a(7) = 2 (see also A024787).
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Column 3 of A194812.

Cf. A135010, A138121, A174455, A182703, A182712, A182714, A240056.

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0]
    elif i<2 then [0, 0]
    else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
         [g[1]+h[1], g[2]+h[2]+`if`(i=3, h[1], 0)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..70);  # Alois P. Heinz, Mar 18 2012

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 1] < Count[p, 2]], {n, 0, z}] (* Clark Kimberling, Mar 31 2014 *)
b[n_, i_] := b[n, i] = Module[{g, h}, If[n == 0, {1, 0}, If[i<2, {0, 0}, g = b[n, i-1]; h = If[i>n, {0, 0}, b[n-i, i]]; Join[g[[1]] + h[[1]], g[[2]] + h[[2]] + If[i == 3, h[[1]], 0]]]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Nov 30 2015, after Alois P. Heinz *)
Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], 3], {n, 51}] (* Robert Price, May 15 2020 *)

A182713 = lambda n: sum(list(p).count(3) for p in Partitions(n) if 1 not in p) # D. S. McNeil, Nov 29 2010

It appears that A000041(n) = a(n+1) + a(n+2) + a(n+3), n >= 0. - Omar E. Pol, Feb 04 2012

a(n) ~ A000041(n)/3 ~ exp(Pi*sqrt(2*n/3)) / (12*sqrt(3)*n). - Vaclav Kotesovec, Jan 03 2019

A193827 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, 2, 4, 3, 3, 2, 5, 4, 2, 2, 4, 3, 2, 2, 3, 6, 5, 4, 3, 2, 5, 4, 2, 2, 3, 7, 3, 3, 6, 5, 2, 2, 4, 3, 2, 2, 3, 6, 5, 4, 2, 2, 2, 2, 3, 4, 8, 4, 3, 7, 6, 5, 3, 2, 5, 4, 2, 2, 3, 7, 3, 3, 6, 5, 2, 2, 2, 2, 3, 3, 4, 9, 5, 4, 3, 4, 8, 7, 6

Offset: 0

Omar E. Pol, Aug 12 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 A186114. For another version see A183152.

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

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

Cf. A135010, A138121.

A194714 Sum of all odd-indexed parts minus the sum of all even-indexed parts of all partitions of n, with the parts written in nondecreasing order.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 14, 18, 26, 32, 48, 57, 82, 102, 138, 169, 230, 278, 370, 450, 584, 709, 914, 1102, 1400, 1692, 2124, 2555, 3186, 3818, 4720, 5649, 6926, 8269, 10078, 11989, 14526, 17249, 20782, 24603, 29508, 34843, 41600, 49008, 58258, 68468, 81098

Offset: 1

Omar E. Pol, Feb 12 2012

nonn

It appears that A066897 is also another version of this sequence but with the parts written in nonincreasing order.

			a(6) = 37 - 29 = 8 because the partitions of 6 written in nondecreasing order are
.
.   6                        =  6
.   3  - 3                   =  0
.   2  - 4                   = -2
.   2  - 2  + 2              =  2
.   1  - 5                   = -4
.   1  - 2  + 3              =  2
.   1  - 1  + 4              =  4
.   1  - 1  + 2 - 2          =  0
.   1  - 1  + 1 - 3          = -2
.   1  - 1  + 1 - 1 + 2      =  2
.   1  - 1  + 1 - 1 + 1 - 1  =  0
----------------------------------
.  20 - 21 + 14 - 7 + 3 - 1  =  8

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A066897, A135010, A138121, A141285, A194853.

More terms from Alois P. Heinz, Feb 12 2012

A207034 Sum of all parts minus the number of parts of the n-th partition in the list of colexicographically ordered partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 4, 5, 5, 6, 6, 6, 7, 5, 6, 6, 7, 6, 7, 7, 8, 5, 6, 6, 7, 7, 7, 8, 7, 8, 8, 8, 9, 6, 7, 7, 8, 7, 8, 8, 9, 8, 8, 9, 9, 9, 10, 6, 7, 7, 8, 8, 8, 9, 8, 9, 9, 9, 10, 8, 9, 9, 10, 9, 10, 10, 10, 11, 7, 8, 8, 9, 8, 9

Offset: 1

Omar E. Pol, Feb 20 2012

a(n) is also the column number in which is located the part of size 1 in the n-th zone of the tail of the last section of the set of partitions of k in colexicographic order, minus the column number in which is located the part of size 1 in the first row of the same tail, when k -> infinity (see example). For the definition of "section" see A135010.

			Illustration of initial terms, n = 1..15. Consider the last 15 rows of the tail of the last section of the set of partitions in colexicographic order of any integer >= 8. The tail contains at least A000041(8-1) = 15 parts of size 1. a(n) is also the number of dots in the n-th row of the diagram.
----------------------------------
n      Tail                  a(n)
----------------------------------
15        1 . . . . . .       6
14          1 . . . . .       5
13          1 . . . . .       5
12            1 . . . .       4
11          1 . . . . .       5
10            1 . . . .       4
9             1 . . . .       4
8               1 . . .       3
7             1 . . . .       4
6               1 . . .       3
5               1 . . .       3
4                 1 . .       2
3                 1 . .       2
2                   1 .       1
1                     1       0
----------------------------------
Written as a triangle:
0;
1;
2;
2,3;
3,4;
3,4,4,5;
4,5,5,6;
4,5,5,6,6,6,7;
5,6,6,7,6,7,7,8;
5,6,6,7,7,7,8,7,8,8,8,9;
6,7,7,8,7,8,8,9,8,8,9,9,9,10;
6,7,7,8,8,8,9,8,9,9,9,10,8,9,9,10,9,10,10,10,11;
...
Consider a matrix [j X A000041(j)] in which the rows represent the partitions of j in colexicographic order (see A211992). Every part of every partition is located in a cell of the matrix. We can see that a(n) is the number of empty cells in row n for any integer j, if A000041(j) >= n. The number of empty cells in row n equals the sum of all parts minus the number of parts in the n-th partition of j.
Illustration of initial terms. The smallest part of every partition is located in the last column of the matrix.
---------------------------------------------------------
.   j: 1    2       3         4           5             6
n a(n)
---------------------------------------------------------
1  0 | 1  1 1   1 1 1   1 1 1 1   1 1 1 1 1   1 1 1 1 1 1
2  1 |    . 2   . 2 1   . 2 1 1   . 2 1 1 1   . 2 1 1 1 1
3  2 |          . . 3   . . 3 1   . . 3 1 1   . . 3 1 1 1
4  2 |                  . . 2 2   . . 2 2 1   . . 2 2 1 1
5  3 |                  . . . 4   . . . 4 1   . . . 4 1 1
6  3 |                            . . . 3 2   . . . 3 2 1
7  4 |                            . . . . 5   . . . . 5 1
8  3 |                                        . . . 2 2 2
9  4 |                                        . . . . 4 2
10 4 |                                        . . . . 3 3
11 5 |                                        . . . . . 6
...
Illustration of initial terms. In this case the largest part of every partition is located in the first column of the matrix.
---------------------------------------------------------
.   j: 1    2       3         4           5             6
n a(n)
---------------------------------------------------------
1  0 | 1  1 1   1 1 1   1 1 1 1   1 1 1 1 1   1 1 1 1 1 1
2  1 |    2 .   2 1 .   2 1 1 .   2 1 1 1 .   2 1 1 1 1 .
3  2 |          3 . .   3 1 . .   3 1 1 . .   3 1 1 1 . .
4  2 |                  2 2 . .   2 2 1 . .   2 2 1 1 . .
5  3 |                  4 . . .   4 1 . . .   4 1 1 . . .
6  3 |                            3 2 . . .   3 2 1 . . .
7  4 |                            5 . . . .   5 1 . . . .
8  3 |                                        2 2 2 . . .
9  4 |                                        4 2 . . . .
10 4 |                                        3 3 . . . .
11 5 |                                        6 . . . . .
...

Alois P. Heinz, Table of n, a(n) for n = 1..10143

Row r has length A187219(r). Partial sums give A207038. Row sums give A207035. Right border gives A001477. Where records occur give A000041 without repetitions.

Cf. A135010, A138121, A141285, A182703, A194548, A196087, A207031, A207032, A207035, A211992, A228716, A230440.

a(n) = t(n) - A194548(n), if n >= 2, where t(n) is the n-th element of the following sequence: triangle read by rows in which row n lists n repeated k times, where k = A187219(n).
a(n) = A000120(A194602(n-1)) = A000120(A228354(n)-1).
a(n) = i - A193173(i,n), i >= 1, 1<=n<=A000041(i).

cp's OEIS Frontend

A211994 A list of ordered partitions of the positive integers.

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A340031 Irregular triangle read by rows T(n,k) in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the j-th row of triangle A127093, where j = n - m + 1 and 1 <= m <= n.

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A100818 For a given unrestricted partition pi, let P(pi)=lambda(pi), if mu(pi)=0. If mu(pi)>0 then let P(pi)=nu(pi), where nu(pi) is the number of parts of pi greater than mu(pi), mu(pi) is the number of ones in pi and lambda(pi) is the largest part of pi.

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A138151 Irregular triangle read by rows in which rows 1..n (when read together) list all the parts in the partitions of n and row n starts with the partitions of n that do not contain 1 as a part (in the order used for A080577).

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A139100 Triangle read by rows: row n lists all partitions of n in the order produced by the shell model of partitions A138151.

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A168018 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n.

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A182713 Number of 3's in the last section of the set of partitions of n.

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A193827 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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