A135010 -id:A135010 - cp's OEIS frontend

A182703 Triangle read by rows: T(n,k) = number of occurrences of k in the last section of the set of partitions of n.

1, 1, 1, 2, 0, 1, 3, 2, 0, 1, 5, 1, 1, 0, 1, 7, 4, 2, 1, 0, 1, 11, 3, 2, 1, 1, 0, 1, 15, 8, 3, 3, 1, 1, 0, 1, 22, 7, 6, 2, 2, 1, 1, 0, 1, 30, 15, 6, 5, 3, 2, 1, 1, 0, 1, 42, 15, 10, 5, 4, 2, 2, 1, 1, 0, 1, 56, 27, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1

Offset: 1

Omar E. Pol, Nov 28 2010

For the definition of "section" of the set of partitions of n see A135010.
Also, column 1 gives the number of partitions of n-1. For k >= 2, row n lists the number of k's in all partitions of n that do not contain 1 as a part.
From Omar E. Pol, Feb 12 2012: (Start)
It appears that reversed rows converge to A002865.
It appears that row n is also the base of an isosceles triangle in which the column sums give the partition numbers A000041 in descending order starting with p(n-1) = A000041(n-1). Example for n = 7:
.
.         1,
.      1, 0, 1,
.   4, 2, 1, 0, 1,
11, 3, 2, 1, 1, 0, 1,
---------------------
11, 7, 5, 3, 2, 1, 1,
.
It appears that in row n starts an infinite trapezoid in which column sums always give the number of partitions of n-1. Example for n = 7:
.
11, 3, 2, 1, 1, 0, 1,
.   8, 3, 3, 1, 1, 0, 1,
.      6, 2, 2, 1, 1, 0, 1,
.         5, 3, 2, 1, 1, 0, 1,
.            4, 2, 2, 1, 1, 0, 1,
.               5, 2, 2, 1, 1, 0,...
.                  4, 2, 2, 1, 1,...
.                     4, 2, 2, 1,...
.                        4, 2, 2,...
.                           4, 2,...
.                              4,...
.
The sum of any column is always p(7-1) = p(6) = A000041(6) = 11.
It appears that the first term of row n is one of the vertices of an infinite isosceles triangle in which column sums give the partition numbers A000041 in ascending order starting with p(n-1) = A000041(n-1). Example for n = 7:
11,
.    8,
.    7,  6,
.        6,  5,
.       10,  5, ...
.           10, ...
.           10, ...
-------------------
11, 15, 22, 30, ...
(End)
It appears that row n lists the first differences of the row n of triangle A207031 together with 1 (as the final term of row n). - Omar E. Pol, Feb 26 2012
More generally T(n,k) is the number of occurrences of k in the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Oct 21 2013

			Illustration of three arrangements of the last section of the set of partitions of 7, or more generally the 7th section of the set of partitions of any integer >= 7:
.                                        _ _ _ _ _ _ _
.     (7)                    (7)        |_ _ _ _      |
.     (4+3)                (4+3)        |_ _ _ _|_    |
.     (5+2)                (5+2)        |_ _ _    |   |
.     (3+2+2)            (3+2+2)        |_ _ _|_ _|_  |
.       (1)                  (1)                    | |
.         (1)                (1)                    | |
.         (1)                (1)                    | |
.           (1)              (1)                    | |
.         (1)                (1)                    | |
.           (1)              (1)                    | |
.           (1)              (1)                    | |
.             (1)            (1)                    | |
.             (1)            (1)                    | |
.               (1)          (1)                    | |
.                 (1)        (1)                    |_|
.    ----------------
.     19,8,5,3,2,1,1 --> Row 7 of triangle A207031.
.      |/|/|/|/|/|/|
.     11,3,2,1,1,0,1 --> Row 7 of this triangle.
.
Note that the "head" of the last section is formed by the partitions of 7 that do not contain 1 as a part. The "tail" is formed by A000041(7-1) parts of size 1. The number of rows (or zones) is A000041(7) = 15. The last section of the set of partitions of 7 contains eleven 1's, three 2's, two 3's, one 4, one 5, there are no 6's and it contains one 7. So, for k = 1..7, row 7 gives: 11, 3, 2, 1, 1, 0, 1.
Triangle begins:
   1;
   1,  1;
   2,  0,  1;
   3,  2,  0,  1;
   5,  1,  1,  0, 1;
   7,  4,  2,  1, 0, 1;
  11,  3,  2,  1, 1, 0, 1;
  15,  8,  3,  3, 1, 1, 0, 1;
  22,  7,  6,  2, 2, 1, 1, 0, 1;
  30, 15,  6,  5, 3, 2, 1, 1, 0, 1;
  42, 15, 10,  5, 4, 2, 2, 1, 1, 0, 1;
  56, 27, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Omar E. Pol, Illustration of the section model of partitions, Figure 1 (n = 1..6), Figure 2 (2D view, n = 1..10), Figure 3 (3D view, n = 1..9)

Row sums give A138137. Where records occur is A134869.
Columns 1-10: A000041, A182712-A182714, A206555-A206560.
Sub-triangles (1-11): A023531, A129186, A194702-A194710
Cf. A066633, A135010, A182742, A182743, A194812, A206563, A207031, A207032, A206437, A211025.

p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
b:= proc(n,i) option remember; local g;
      if n=0        then [1]
    elif n<2 or i<2 then [0]
    else g:=   `if`(i>n, [0],  b(n-i, i));
         p(p([0$j=2..i, g[1]], b(n, i-1)), g)
      fi
    end:
h:= proc(n) option remember;
      `if`(n=0, 1, b(n, n)[1]+h(n-1))
    end:
T:= proc(n) h(n-1), b(n, n)[2..n][] end:
seq(T(n), n=1..20);  # Alois P. Heinz, Feb 19 2012

p[f_, g_] := Plus @@ PadRight[{f, g}]; b[n_, i_] := b[n, i] = Module[{g}, Which[n == 0, {1}, n<2 || i<2, {0}, True, g = If [i>n, {0}, b[n-i, i]]; p[p[Append[Array[0&, i-1], g[[1]]], b[n, i-1]], g]]]; h[n_] := h[n] = If[n == 0, 1, b[n, n][[1]] + h[n-1]]; t[n_] := {h[n-1], Sequence @@ b[n, n][[2 ;; n]]}; Table[t[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 16 2014, after Alois P. Heinz's Maple code *)
Table[{PartitionsP[n-1]}~Join~Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], k], {k,2,n}], {n,1,12}]  // Flatten (* Robert Price, May 15 2020 *)

It appears that T(n,k) = A207032(n,k) - A207032(n,k+2). - Omar E. Pol, Feb 26 2012

A206437 Triangle read by rows: T(j,k) is the k-th part of the j-th region of the set of partitions of n, if 1 <= j <= A000041(n).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 14 2012

Here the j-th "region" of the set of partitions of n (or more simply the j-th "region" of n) is defined to be the first h elements of the sequence formed by the smallest parts in nonincreasing order of the partitions of the largest part of the j-th partition of n, with the list of partitions in colexicographic order, where h = j - i, and i is the index of the previous partition of n whose largest part is greater than the largest part of the j-th partition of n, or i = 0 if such previous largest part does not exist. The largest part of the j-th region of n is A141285(j) and the number of parts is h = A194446(j).
Some properties of the regions of n:
- The number of regions of n equals the number of partitions of n (see A000041).
- The set of regions of n contain the sets of regions of all positive integers previous to n.
- The first j regions of n are also first j regions of all integers greater than n.
- The sums of all largest parts of all regions of n equals the total number of parts of all regions of n. See A006128(n).
- If T(j,1) is a record in the sequence then the leading diagonals of triangle formed by the first j rows give the partitions of n (see example).
- The rank of a region is the largest part minus the number of parts (see A194447).
- The sum of all ranks of the regions of n is equal to zero.
How to make a diagram of the regions and partitions of n: in the first quadrant of the square grid we draw a horizontal line {[0, 0],[n, 0]} of length n. Then we draw a vertical line {[0, 0],[0, p(n)]} of length p(n) where p(n) is the number of partitions of n. Then, for j = 1..p(n), we draw a horizontal line {[0, j],[g, j]} where g = A141285(j) is the largest part of the j-th partition of n, with the list of partitions in colexicographic order. Then, for n = 1 .. p(n), we draw a vertical line from the point [g,j] down to intercept the next segment in a lower row. So we have a number of closed regions. Then we divide each region of n in horizontal rectangles with shorter sides = 1. We can see that in the original rectangle of area n*p(n) each row contains a set of rectangles whose areas are equal to the parts of one of the partitions of n. Then each region of n is labeled according to the position of its largest part on axis "y". Note that each region of n is similar to a mirror version of the Young diagram of one of the partitions of s, where s is the sum of all parts of the region. See the illustrations of the seven regions of 5 in the Links section.
Note that if row j of triangle contains parts of size 1 then the parts of row j are the smallest parts of all partitions of T(j,1), (see A046746), and also T(j,1) is a record in the sequence and also j is the number of partitions of T(j,1), (see A000041). Otherwise, if row j does not contain parts of size 1 then the parts of row j are the emergent parts of the next record in the sequence (see A183152). Row j is also the partition of A186412(j).
Also triangle read by rows in which row r lists the parts of the last section of the set of partitions of r, ordered by regions, such that the previous parts to the part of size r are the emergent parts of the partitions of r (see A138152) and the rest are the smallest parts of the partitions of r (see example). - Omar E. Pol, Apr 28 2012

			-------------------------------------------
  Region j   Triangle of parts
-------------------------------------------
  1          1;
  2          2,1;
  3          3,1,1;
  4          2;
  5          4,2,1,1,1;
  6          3;
  7          5,2,1,1,1,1,1;
  8          2;
  9          4,2;
  10         3;
  11         6,3,2,2,1,1,1,1,1,1,1;
  12         3;
  13         5,2;
  14         4;
  15         7,3,2,2,1,1,1,1,1,1,1,1,1,1,1;
.
The rotated triangle shows each row as a partition:
                             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
.
Alternative interpretation of this sequence:
Triangle read by rows in which row r lists the parts of the last section of the set of partitions of r ordered by regions (see comments):
   [1];
   [2,1];
   [3,1,1];
   [2],[4,2,1,1,1];
   [3],[5,2,1,1,1,1,1];
   [2],[4,2],[3],[6,3,2,2,1,1,1,1,1,1,1];
   [3],[5,2],[4],[7,3,2,2,1,1,1,1,1,1,1,1,1,1,1];

Robert Price, Table of n, a(n) for n = 1..321, first 75 regions.
Omar E. Pol, Illustration of the seven regions of 5
Omar E. Pol, Illustration of initial terms, regions = 1..77 (2D view)
Omar E. Pol, Illustration of initial terms, regions = 1..30 (3D view)
Omar E. Pol, Visualization of regions in a diagram for A006128
Robert Price, Mathematica program to draw diagram up to n=28

Positive integers in A193870. Column 1 is A141285. Row j has length A194446(j). Row sums give A186412. Records are A000027.

Cf. A000041, A046746, A135010, A138121, A182699, A182703, A182709, A183152, A186114, A187219, A194436-A194439, A194447, A194448, A196025, A198381.

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0, 2];
reg = {}; l = {};
For[j = 1, j <= 22, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  AppendTo[reg, Take[Reverse[First /@ lex[mx]], j - i]];
  ];
Flatten@reg  (* Robert Price, Apr 21 2020, revised Jul 24 2020 *)

Further edited by Omar E. Pol, Mar 31 2012, Jan 27 2013
Minor edits by Omar E. Pol, Apr 23 2020
Comments corrected (following a suggestion from Peter Munn) by Omar E. Pol, Jul 20 2025

A141285 Largest part of the n-th partition of j in the list of colexicographically ordered partitions of j, if 1 <= n <= A000041(j).

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 5, 2, 4, 3, 6, 3, 5, 4, 7, 2, 4, 3, 6, 5, 4, 8, 3, 5, 4, 7, 3, 6, 5, 9, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8, 7, 6, 11, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 6, 5, 9, 4, 8, 7, 6, 12

Offset: 1

Omar E. Pol, Aug 01 2008

Also largest part of the n-th region of the set of partitions of j, if 1 <= n <= A000041(j). For the definition of "region of the set of partitions of j" see A206437.
Also triangle read by rows: T(j,k) is the largest part of the k-th region in the last section of the set of partitions of j.
For row n >= 2 the rows of triangle are also the branches of a tree which is a projection of a three-dimensional structure of the section model of partitions of A135010, version tree. The branches of even rows give A182730. The branches of odd rows give A182731. Note that each column contains parts of the same size. It appears that the structure of A135010 is a periodic table of integer partitions. See also A210979 and A210980.
Also column 1 of: A193870, A206437, A210941, A210942, A210943. - Omar E. Pol, Sep 01 2013
Also row lengths of A211009. - Omar E. Pol, Feb 06 2014

			Written as a triangle T(j,k) the sequence begins:
  1;
  2;
  3;
  2, 4;
  3, 5;
  2, 4, 3, 6;
  3, 5, 4, 7;
  2, 4, 3, 6, 5, 4, 8;
  3, 5, 4, 7, 3, 6, 5, 9;
  2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10;
  3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8,  7, 6, 11;
  ...
  ------------------------------------------
  n  A000041                a(n)
  ------------------------------------------
   1 = p(1)                   1
   2 = p(2)                 2 .
   3 = p(3)                   . 3
   4                        2 .
   5 = p(4)               4   .
   6                          . 3
   7 = p(5)                   .   5
   8                        2 .
   9                      4   .
  10                    3     .
  11 = p(6)           6       .
  12                          . 3
  13                          .   5
  14                          .     4
  15 = p(7)                   .       7
  ...
From _Omar E. Pol_, Aug 22 2013: (Start)
Illustration of initial terms (n = 1..11) in three ways: as the largest parts of the partitions of 6 (see A026792), also as the largest parts of the regions of the diagram, also as the diagonal of triangle. By definition of "region" the largest part of the n-th region is also the largest part of the n-th partition (see below):
  --------------------------------------------------------
  .                  Diagram         Triangle in which
  Partitions       of regions       rows are partitions
  of 6           and partitions   and columns are regions
  --------------------------------------------------------
  .                _ _ _ _ _ _
  6                _ _ _      |                         6
  3+3              _ _ _|_    |                       3 3
  4+2              _ _    |   |                     4   2
  2+2+2            _ _|_ _|_  |                   2 2   2
  5+1              _ _ _    | |                 5       1
  3+2+1            _ _ _|_  | |               3 1       1
  4+1+1            _ _    | | |             4   1       1
  2+2+1+1          _ _|_  | | |           2 2   1       1
  3+1+1+1          _ _  | | | |         3   1   1       1
  2+1+1+1+1        _  | | | | |       2 1   1   1       1
  1+1+1+1+1+1       | | | | | |     1 1 1   1   1       1
  ...
The equivalent sequence for compositions is A001511. Explanation: for the positive integer j the diagram of regions of the set of compositions of j has 2^(j-1) regions. The largest part of the n-th region is A001511(n). The number of parts is A006519(n). On the other hand the diagram of regions of the set of partitions of j has A000041(j) regions. The largest part of the n-th region is a(n) = A001511(A228354(n)). The number of parts is A194446(n). Both diagrams have j sections. The diagram for partitions can be interpreted as one of the three views of a three dimensional diagram of compositions in which the rows of partitions are in orthogonal direction to the rest. For the first five sections of the diagrams see below:
  --------------------------------------------------------
  .          Diagram                           Diagram
  .         of regions                        of regions
  .      and compositions                   and partitions
  ---------------------------------------------------------
  .      j = 1 2 3 4 5                     j = 1 2 3 4 5
  ---------------------------------------------------------
   n  A001511                    A228354  a(n)
  ---------------------------------------------------------
   1   1     _| | | | | ............ 1    1    _| | | | |
   2   2     _ _| | | | ............ 2    2    _ _| | | |
   3   1     _|   | | |    ......... 4    3    _ _ _| | |
   4   3     _ _ _| | | ../  ....... 6    2    _ _|   | |
   5   1     _| |   | |    / ....... 8    4    _ _ _ _| |
   6   2     _ _|   | | ../ /   .... 12   3    _ _ _|   |
   7   1     _|     | |    /   /   . 16   5    _ _ _ _ _|
   8   4     _ _ _ _| | ../   /   /
   9   1     _| | |   |      /   /
  10   2     _ _| |   |     /   /
  11   1     _|   |   |    /   /
  12   3     _ _ _|   | ../   /
  13   1     _| |     |      /
  14   2     _ _|     |     /
  15   1     _|       |    /
  16   5     _ _ _ _ _| ../
  ...
Also we can draw an infinite Dyck path in which the n-th odd-indexed line segment has a(n) up-steps and the n-th even-indexed line segment has A194446(n) down-steps. Note that the height of the n-th largest peak between two successive valleys at height 0 is also the partition number A000041(n). See below:
.                                 5
.                                 /\                 3
.                   4            /  \           4    /\
.                   /\          /    \          /\  /
.         3        /  \     3  /      \        /  \/
.    2    /\   2  /    \    /\/        \   2  /
. 1  /\  /  \  /\/      \  /            \  /\/
. /\/  \/    \/          \/              \/
.
.(End)

Alois P. Heinz, Table of n, a(n) for n = 1..4000
Omar E. Pol, Illustration of initial terms
Omar E. Pol, Illustration of the seven regions of 5

Where records occur give A000041, n>=1. Column 1 is A158478. Row j has length A187219(j). Row sums give A138137. Right border gives A000027.

Cf. A000041, A135010, A182730, A182731, A182732, A182733, A182982, A182983, A182703, A193870, A194446, A194447, A194550, A206437, A210979, A210980, A211978, A220517, A225600, A225610.

Last/@DeleteCases[DeleteCases[Sort@PadRight[Reverse/@IntegerPartitions[13]], x_ /; x == 0, 2], {}] (* updated _Robert Price, May 15 2020 *)

a(n) = A001511(A228354(n)). - Omar E. Pol, Aug 22 2013

Edited by Omar E. Pol, Nov 28 2010

Better definition and edited by Omar E. Pol, Oct 17 2013

A138137 First differences of A006128.

Original entry on oeis.org

1, 2, 3, 6, 8, 15, 19, 32, 42, 64, 83, 124, 157, 224, 288, 395, 502, 679, 854, 1132, 1422, 1847, 2307, 2968, 3677, 4671, 5772, 7251, 8908, 11110, 13572, 16792, 20439, 25096, 30414, 37138, 44798, 54389, 65386, 78959, 94558, 113687, 135646, 162375, 193133

Offset: 1

Omar E. Pol, Mar 18 2008

Number of parts in the last section of the set of partitions of n (see A135010, A138121).
Sum of largest parts in all partitions in the head of the last section of the set of partitions of n. - Omar E. Pol, Nov 07 2011
From Omar E. Pol, Feb 16 2021: (Start)
Convolution of A341062 and A000041.
Convolution of A000005 and A002865.
a(n) is also the total number of parts in the n-th section of the set of partitions of any positive integer >= n.
a(n) is also the total number of divisors of all terms in the n-th row of triangle A336811. These divisors are also all parts in the last section of the set of partitions of n. (End)

			From _Omar E. Pol_, Feb 19 2012: (Start)
Illustration of initial terms (n = 1..6) as sums of the first columns from the last sections of the first six natural numbers (or from the first six sections of 6):
.                                           6
.                                           3+3
.                                           4+2
.                                           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
. --- ----- ------- --------- ----------- --------------
.  1,  2,    3,      6,        8,          15,
...
Also, we can see that the sequence gives the number of parts in each section. For the number of odd/even parts (and more) see A207031, A207032 and also A206563. (End)
From _Omar E. Pol_, Aug 16 2013: (Start)
The geometric model looks like this:
.                                           _ _ _ _ _ _
.                                          |_ _ _ _ _ _|
.                                          |_ _ _|_ _ _|
.                                          |_ _ _ _|_ _|
.                               _ _ _ _ _  |_ _|_ _|_ _|
.                              |_ _ _ _ _|           |_|
.                     _ _ _ _  |_ _ _|_ _|           |_|
.                    |_ _ _ _|         |_|           |_|
.             _ _ _  |_ _|_ _|         |_|           |_|
.       _ _  |_ _ _|       |_|         |_|           |_|
.   _  |_ _|     |_|       |_|         |_|           |_|
.  |_|   |_|     |_|       |_|         |_|           |_|
.
.   1    2      3        6          8           15
.
(End)
On the other hand for n = 6 the 6th row of triangle A336811 is [6, 4, 3, 2, 2, 1, 1] and the divisors of these terms are [1, 2, 3, 6], [1, 2, 4], [1, 3], [1, 2], [1, 2], [1], [1]. There are 15 divisors so a(6) = 15. - _Omar E. Pol_, Jul 27 2021

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

Column 1 of A207031.

Cf. A000005, A000041, A002865, A006128, A135010, A138121, A182703, A336811, A336812, A338156, A340035, A341062.

b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    elif i>n then b(n, i-1)
    else f:= b(n, i-1); g:= b(n-i, i);
         [f[1]+g[1], f[2]+g[2] +g[1]]
      fi
    end:
a:= n-> b(n, n)[2] -b(n-1, n-1)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 19 2012

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n == 0, {1, 0}, i<1, {0, 0}, i>n, b[n, i-1], True, f = b[n, i-1]; g = b[n-i, i]; {f[[1]]+g[[1]], f[[2]]+g[[2]]+g[[1]]}]]; a[n_] := b[n, n][[2]]-b[n-1, n-1][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
Table[PartitionsP[n - 1] + Length@Flatten@Select[IntegerPartitions[n], FreeQ[#, 1] &], {n, 1, 45}] (* Robert Price, May 01 2020 *)

a(n) = A006128(n) - A006128(n-1).
a(n) = A000041(n-1) + A138135(n). - Omar E. Pol, Nov 07 2011
a(n) ~ exp(Pi*sqrt(2*n/3)) * (2*gamma + log(6*n/Pi^2)) / (8*sqrt(3)*n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 21 2016
G.f.: Sum_{i>=1} i*x^i * Product_{j=2..i} 1/(1 - x^j). - Ilya Gutkovskiy, Apr 04 2017

A225620 Indices of partitions in the table of compositions of A228351.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 10, 12, 14, 15, 16, 20, 24, 26, 28, 30, 31, 32, 36, 40, 42, 48, 52, 56, 58, 60, 62, 63, 64, 72, 80, 84, 96, 100, 104, 106, 112, 116, 120, 122, 124, 126, 127, 128, 136, 144, 160, 164, 168, 170, 192, 200, 208, 212, 224, 228, 232, 234, 240, 244, 248, 250, 252, 254, 255

Offset: 1

Omar E. Pol, Aug 03 2013

Also triangle read by rows in which T(n,k) is the decimal representation of a binary number whose mirror represents the k-th partition of n according with the list of juxtaposed reverse-lexicographically ordered partitions of the positive integers (A026792).
In order to construct this sequence as a triangle we use the following rules:
- In the list of A026792 we replace each part of size j of the k-th partition of n by concatenation of j - 1 zeros and only one 1.
- Then replace this new set of parts by the concatenation of its parts.
- Then replace this string by its mirror version which is a binary number.
T(n,k) is the decimal value of this binary number, which represents the k-th partition of n (see example).
The partitions of n are represented by a subsequence with A000041(n) integers starting with 2^(n-1) and ending with 2^n - 1, n >= 1. The odd numbers of the sequence are in A000225.
First differs from A065609 at a(23).
Conjecture: this sequence is a sorted version of b(n) where b(2^k) = 2^k for k >= 0, b(n) = A080100(n)*(2*b(A053645(n)) + 1) otherwise. - Mikhail Kurkov, Oct 21 2023

			T(6,8) = 58 because 58 in base 2 is 111010 whose mirror is 010111 which is the concatenation of 01, 01, 1, 1, whose number of digits are 2, 2, 1, 1, which are also the 8th partition of 6.
Illustration of initial terms:
The sequence represents a table of partitions (see below):
--------------------------------------------------------
.            Binary                        Partitions
n  k  T(n,k) number  Mirror   Diagram       (A026792)
.                                          1 2 3 4 5 6
--------------------------------------------------------
.                             _
1  1     1       1    1        |           1,
.                             _ _
1  1     2      10    01      _  |           2,
2  2     3      11    11       | |         1,1,
.                             _ _ _
3  1     4     100    001     _ _  |           3,
3  2     6     110    011     _  | |         2,1,
3  3     7     111    111      | | |       1,1,1,
.                             _ _ _ _
4  1     8    1000    0001    _ _    |           4,
4  2    10    1010    0101    _ _|_  |         2,2,
4  3    12    1100    0011    _ _  | |         3,1,
4  4    14    1110    0111    _  | | |       2,1,1,
4  5    15    1111    1111     | | | |     1,1,1,1,
.                             _ _ _ _ _
5  1    16   10000    00001   _ _ _    |           5,
5  2    20   10100    00101   _ _ _|_  |         3,2,
5  3    24   11000    00011   _ _    | |         4,1,
5  4    26   11010    01011   _ _|_  | |       2,2,1,
5  5    28   11100    00111   _ _  | | |       3,1,1,
5  6    30   11110    01111   _  | | | |     2,1,1,1,
5  7    31   11111    11111    | | | | |   1,1,1,1,1,
.                             _ _ _ _ _ _
6  1    32  100000    000001  _ _ _      |           6
6  2    36  100100    001001  _ _ _|_    |         3,3,
6  3    40  101000    000101  _ _    |   |         4,2,
6  4    42  101010    010101  _ _|_ _|_  |       2,2,2,
6  5    48  110000    000011  _ _ _    | |         5,1,
6  6    52  110100    001011  _ _ _|_  | |       3,2,1,
6  7    56  111000    000111  _ _    | | |       4,1,1,
6  8    58  111010    010111  _ _|_  | | |     2,2,1,1,
6  9    60  111100    001111  _ _  | | | |     3,1,1,1,
6  10   62  111110    011111  _  | | | | |   2,1,1,1,1,
6  11   63  111111    111111   | | | | | | 1,1,1,1,1,1,
.
Triangle begins:
  1;
  2,   3;
  4,   6,  7;
  8,  10, 12, 14, 15;
  16, 20, 24, 26, 28, 30, 31;
  32, 36, 40, 42, 48, 52, 56, 58, 60, 62, 63;
  ...
From _Gus Wiseman_, Apr 01 2020: (Start)
Using the encoding of A066099, this sequence ranks all finite nonempty multisets, as follows.
   1: {1}
   2: {2}
   3: {1,1}
   4: {3}
   6: {1,2}
   7: {1,1,1}
   8: {4}
  10: {2,2}
  12: {1,3}
  14: {1,1,2}
  15: {1,1,1,1}
  16: {5}
  20: {2,3}
  24: {1,4}
  26: {1,2,2}
  28: {1,1,3}
  30: {1,1,1,2}
  31: {1,1,1,1,1}
(End)

Column 1 is A000079. Row n has length A000041(n). Right border gives A000225.
Cf. A026792, A065609, A135010, A141285, A186114, A194446, A194546, A206437, A207779, A211978, A225600, A225610, A228351.
The case covering an initial interval is A333379 or A333380.
All of the following pertain to compositions in the order of A066099.
- The weakly increasing version is this sequence.
- The weakly decreasing version is A114994.
- The strictly increasing version is A333255.
- The strictly decreasing version is A333256.
- The unequal version is A233564.
- The equal version is A272919.
- The case covering an initial interval is A333217.
- Initial intervals are ranked by A164894.
- Reversed initial intervals are ranked by A246534.
Cf. A000120, A029931, A048793, A070939, A124766, A233249, A333219, A333220.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],LessEqual@@stc[#]&] (* Gus Wiseman, Apr 01 2020 *)

Conjecture: a(A000070(m) - k) = 2^m - A228354(k) for m > 0, 0 < k <= A000041(m). - Mikhail Kurkov, Oct 20 2023

A187219 Number of partitions of n that do not contain parts less than the smallest part of the partitions of n-1.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 41, 55, 66, 88, 105, 137, 165, 210, 253, 320, 383, 478, 574, 708, 847, 1039, 1238, 1507, 1794, 2167, 2573, 3094, 3660, 4378, 5170, 6153, 7245, 8591, 10087, 11914, 13959, 16424, 19196, 22519, 26252, 30701, 35717

Offset: 1

Omar E. Pol, Dec 09 2011

nonn

Essentially the same as A002865, but here a(1) = 1 not 0.
Also number of regions in the last section of the set of partitions of n.
Also number of partitions of n+k that are formed by k+1 sections, k >= 0 (Cf. A194799). - Omar E. Pol, Jan 30 2012
For the definition of region see A206437. - Omar E. Pol, Aug 13 2013
Partial sums give A000041, n >= 1. - Omar E. Pol, Sep 04 2013
Also the number of partitions of n with no parts greater than the number of ones. - Spencer Miller, Jan 28 2023

			From _Omar E. Pol_, Aug 13 2013: (Start)
Illustration of initial terms as number of regions:
.                                           _ _ _ _ _ _
.                                          |_ _ _      |
.                                          |_ _ _|_    |
.                                          |_ _    |   |
.                               _ _ _ _ _  |_ _|_ _|_  |
.                              |_ _ _    |           | |
.                     _ _ _ _  |_ _ _|_  |           | |
.                    |_ _    |         | |           | |
.             _ _ _  |_ _|_  |         | |           | |
.       _ _  |_ _  |       | |         | |           | |
.   _  |_  |     | |       | |         | |           | |
.  |_|   |_|     |_|       |_|         |_|           |_|
.
.   1     1       1         2           2             4
.
(End)

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
Omar E. Pol, Illustration of the seven regions of 5

Cf. A000041, A002865, A083751, A135010, A186114, A193870, A206437, A225600, A225610.

Join[{1}, Drop[CoefficientList[Series[1 / Product[(1 - x^k)^1, {k, 2, 50}], {x, 0, 50}], x], 2]] (* Vincenzo Librandi, Feb 15 2018 *)
A187219[nmax_]:=Join[{1},Differences[PartitionsP[Range[nmax]]]];
A187219[100] (* Paolo Xausa, Feb 17 2023 *)

a(n) = A083751(n) + 1. - Omar E. Pol, Mar 04 2012

a(n) = A002865(n), if n >= 2. - Omar E. Pol, Aug 13 2013

Better definition from Omar E. Pol, Sep 04 2013

A194446 Number of parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

Original entry on oeis.org

1, 2, 3, 1, 5, 1, 7, 1, 2, 1, 11, 1, 2, 1, 15, 1, 2, 1, 4, 1, 1, 22, 1, 2, 1, 4, 1, 2, 1, 30, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 42, 1, 2, 1, 4, 1, 2, 1, 8, 1, 1, 3, 1, 1, 56, 1, 2, 1, 4, 1, 1, 7, 1, 2, 1, 1, 12, 1, 2, 1, 4, 1, 2, 1, 1, 77, 1, 2, 1

Offset: 1

Omar E. Pol, Nov 26 2011

For the definition of "region" of the set of partitions of j, see A206437.
a(n) is also the number of positive integers in the n-th row of triangle A186114. a(n) is also the number of positive integers in the n-th row of triangle A193870.
Also triangle read by rows: T(j,k) = number of parts in the k-th region of the last section of the set of partitions of j. See example. For more information see A135010.
a(n) is also the length of the n-th vertical line segment in the minimalist diagram of regions and partitions. The length of the n-th horizontal line segment is A141285(n). See also A194447. - Omar E. Pol, Mar 04 2012
From Omar E. Pol, Aug 19 2013: (Start)
In order to construct this sequence with a cellular automaton we use the following rules: We start in the first quadrant of the square grid with no toothpicks. At stage n we place A141285(n) toothpicks of length 1 connected by their endpoints in horizontal direction starting from the point (0, n). Then we place toothpicks of length 1 connected by their endpoints in vertical direction starting from the exposed toothpick endpoint downward up to touch the structure or up to touch the x-axis. a(n) is the number of toothpicks in vertical direction added at n-th stage (see example section and A139250, A225600, A225610).
a(n) is also the length of the n-th descendent line segment in an infinite Dyck path in which the length of the n-th ascendent line segment is A141285(n). See Example section. For more information see A211978, A220517, A225600.
(End)
The equivalent sequence for compositions is A006519. - Omar E. Pol, Aug 22 2013

			Written as an irregular triangle the sequence begins:
  1;
  2;
  3;
  1, 5;
  1, 7;
  1, 2, 1, 11;
  1, 2, 1, 15;
  1, 2, 1,  4, 1, 1, 22;
  1, 2, 1,  4, 1, 2,  1, 30;
  1, 2, 1,  4, 1, 1,  7,  1, 2, 1, 1, 42;
  1, 2, 1,  4, 1, 2,  1,  8, 1, 1, 3,  1, 1, 56;
  1, 2, 1,  4, 1, 1,  7,  1, 2, 1, 1, 12, 1,  2, 1, 4, 1, 2, 1, 1, 77;
  ...
From _Omar E. Pol_, Aug 18 2013: (Start)
Illustration of initial terms (first seven regions):
.                                             _ _ _ _ _
.                                     _ _ _  |_ _ _ _ _|
.                           _ _ _ _  |_ _ _|       |_ _|
.                     _ _  |_ _ _ _|                 |_|
.             _ _ _  |_ _|     |_ _|                 |_|
.       _ _  |_ _ _|             |_|                 |_|
.   _  |_ _|     |_|             |_|                 |_|
.  |_|   |_|     |_|             |_|                 |_|
.
.   1     2       3     1         5       1           7
.
The next figure shows a minimalist diagram of the first seven regions. The n-th horizontal line segment has length A141285(n). a(n) is the length of the n-th vertical line segment, which is the vertical line segment ending in row n (see also A225610).
.      _ _ _ _ _
.  7   _ _ _    |
.  6   _ _ _|_  |
.  5   _ _    | |
.  4   _ _|_  | |
.  3   _ _  | | |
.  2   _  | | | |
.  1    | | | | |
.
.      1 2 3 4 5
.
Illustration of initial terms from an infinite Dyck path in which the length of the n-th ascendent line segment is A141285(n). a(n) is the length of the n-th descendent line segment.
.                                    /\
.                                   /  \
.                      /\          /    \
.                     /  \        /      \
.            /\      /    \    /\/        \
.       /\  /  \  /\/      \  / 1          \
.    /\/  \/    \/ 1        \/              \
.     1   2     3           5               7
.
(End)

Robert Price, Table of n, a(n) for n = 1..5603
Omar E. Pol, Illustration of the seven regions of 5

Row j has length A187219(j). Right border gives A000041, j >= 1. Records give A000041, j >= 1. Row sums give A138137.

Cf. A002865, A006128, A135010, A138121, A186114, A186412, A193870, A194436, A194437, A194438, A194439, A194447.

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
A194446 = {}; l = {};
For[j = 1, j <= 30, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  AppendTo[A194446, j - i];
  ];
A194446   (* Robert Price, Jul 25 2020 *)

a(n) = A141285(n) - A194447(n). - Omar E. Pol, Mar 04 2012

A186114 Triangle of regions and partitions of integers (see Comments lines for definition).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 08 2011

Let r = T(n,k) be a record in the sequence. The consecutive records "r" are the natural numbers A000027. Consider the first n rows; the triangle T(n,k) has the property that the columns, without the zeros, from k..1, are also the partitions of r in juxtaposed reverse-lexicographical order, so k is also A000041(r), the number of partitions of r. Note that a record r is always the final term of a row if such row contains 1’s. The number of positive integer a(1)..r is A006128(r). The sums a(1)..r is A066186(r). Here the set of positive integers in every row (from 1 to n) is called a “region” of r. The number of regions of r equals the number of partitions of r. If T(n,1) = 1 then the row n is formed by the smallest parts, in nondecreasing order, of all partitions of T(n,n).

			Triangle begins:
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,
0, 0, 0, 0, 0, 0, 0, 2,
0, 0, 0, 0, 0, 0, 0, 2, 4,
0, 0, 0, 0, 0, 0, 0, 0, 0, 3,
1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 6
...
The row n = 11 contains the 6th record in the sequence:  a(66) = T(11,11) = 6, then consider the first 11 rows of triangle. Note that the columns, from k = 11..1, without the zeros, are also the 11 partitions of 6 in juxtaposed reverse-lexicographical order: [6], [3, 3], [4, 2], [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]. See A026792.

Robert Price, Table of n, a(n) for n = 1..196878, rows 1-627.
Omar E. Pol, Illustration of the seven regions of 5

Mirror of triangle A193870. Column 1 gives A167392. Right diagonal gives A141285.

Cf. A000041, A135010, A138121, A183152, A186412, A187219, A194436-A194439, A194446-A194448, A206437.

A206437 = Cases[Import["https://oeis.org/A206437/b206437.txt",
     "Table"], {, }][[All, 2]];
A194446 = Cases[Import["https://oeis.org/A194446/b194446.txt",
    "Table"], {, }][[All, 2]];
f[n_] := Module[{v},
   v = Take[A206437, A194446[[n]]];
   A206437 = Drop[A206437, A194446[[n]]];
   Reverse[PadRight[v, n]]];
Table[f[n], {n, PartitionsP[20]}]  // Flatten (* Robert Price, Apr 26 2020 *)

T(n,1) = A167392(n).

T(n,k) = A141285(n), if k = n.

A138879 Sum of all parts of the last section of the set of partitions of n.

Original entry on oeis.org

1, 3, 5, 11, 15, 31, 39, 71, 94, 150, 196, 308, 389, 577, 750, 1056, 1353, 1881, 2380, 3230, 4092, 5412, 6821, 8935, 11150, 14386, 17934, 22834, 28281, 35735, 43982, 55066, 67551, 83821, 102365, 126267, 153397, 188001, 227645, 277305, 334383

Offset: 1

Omar E. Pol, Apr 30 2008

nonn

Row sums of the triangles A135010, A138121, A138151 and others related to the section model of partitions (see A135010 and A138121).
From Omar E. Pol, Jan 20 2021: (Start)
Convolution of A000203 and A002865.
Convolution of A340793 and A000041.
Row sums of triangles A339278, A340426, A340583. (End)
a(n) is also the sum of all divisors of all terms of n-th row of A336811. These divisors are also all parts in the last section of the set of partitions of n. - Omar E. Pol, Jul 27 2021
Row sums of A336812. - Omar E. Pol, Aug 03 2021

			a(6)=31 because the parts of the last section of the set of partitions of 6 are (6), (3,3), (4,2), (2,2,2), (1), (1), (1), (1), (1), (1), (1), so the sum is a(6) = 6 + 3 + 3 + 4 + 2 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 31.
From _Omar E. Pol_, Aug 13 2013: (Start)
Illustration of initial terms:
.                                           _ _ _ _ _ _
.                                          |_ _ _ _ _ _|
.                                          |_ _ _|_ _ _|
.                                          |_ _ _ _|_ _|
.                               _ _ _ _ _  |_ _|_ _|_ _|
.                              |_ _ _ _ _|           |_|
.                     _ _ _ _  |_ _ _|_ _|           |_|
.                    |_ _ _ _|         |_|           |_|
.             _ _ _  |_ _|_ _|         |_|           |_|
.       _ _  |_ _ _|       |_|         |_|           |_|
.   _  |_ _|     |_|       |_|         |_|           |_|
.  |_|   |_|     |_|       |_|         |_|           |_|
.
.   1    3      5        11         15           31
.
(End)
On the other hand for n = 6 the 6th row of triangle A336811 is [6, 4, 3, 2, 2, 1, 1] and the sum of all divisors of these terms is [1 + 2 + 3 + 6] + [1 + 2 + 4] + [1 + 3] + [1 + 2] + [1 + 2] + [1] + [1] = 31, so a(6) = 31. - _Omar E. Pol_, Jul 27 2021

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000041, A000203, A002865, A066186, A133041, A135010, A138121, A138135 - A138138, A138151, A138880, A139100, A237593, A336811, A336812, A338156, A339278, A340035, A340426, A340583, A340793.

A066186 := proc(n) n*combinat[numbpart](n) ; end proc:
A138879 := proc(n) A066186(n)-A066186(n-1) ; end proc:
seq(A138879(n),n=1..80) ; # R. J. Mathar, Jan 27 2011

Table[PartitionsP[n]*n - PartitionsP[n-1]*(n-1), {n, 1, 50}] (* Vaclav Kotesovec, Oct 21 2016 *)

for(n=1, 50, print1(numbpart(n)*n - numbpart(n - 1)*(n - 1),", ")) \\ Indranil Ghosh, Mar 19 2017

from sympy.ntheory import npartitions
print([npartitions(n)*n - npartitions(n - 1)*(n - 1) for n in range(1, 51)]) # Indranil Ghosh, Mar 19 2017

a(n) = A000041(n)*n - A000041(n-1)*(n-1) = A138880(n) + A000041(n-1).
a(n) = A066186(n) - A066186(n-1), for n>=1.
a(n) ~ exp(Pi*sqrt(2*n/3)) * Pi/(12*sqrt(2*n)) * (1 - (72 + 13*Pi^2) / (24*Pi*sqrt(6*n)) + (7/12 + 3/(2*Pi^2) + 217*Pi^2/6912)/n - (15*sqrt(3/2)/(16*Pi) + 115*Pi/(288*sqrt(6)) + 4069*Pi^3/(497664*sqrt(6)))/n^(3/2)). - Vaclav Kotesovec, Oct 21 2016, extended Jul 06 2019
G.f.: x*(1 - x)*f'(x), where f(x) = Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 13 2017

a(34) corrected by R. J. Mathar, Jan 27 2011

A336811 Irregular triangle read by rows T(n,k) in which the length of row n equals the partition number A000041(n-1) and every column k gives the positive integers A000027, with n >= 1 and k >= 1.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 20 2020

In other words: row n lists A028310(n-1) blocks where the m-th block consists of A187219(m) copies of n - m + [m=1], with n >= 1 and m >= 1, where [] is the Iverson bracket. [Corrected by Paolo Xausa, Feb 10 2023]
All divisors of all terms in row n are also all parts in the last section of the set of partitions of n.
Thus all divisors of all terms of the first n rows of triangle are also all parts of all partitions of n. In other words: all divisors of the first A000070(n-1) terms of the sequence are also all parts of all partitions of n. - Omar E. Pol, Jun 19 2021
From Omar E. Pol, Jul 31 2021: (Start)
The number of k's in row n is equal to A002865(n-k), 1 <= k <= n.
The number of terms >= k in row n is equal to A000041(n-k), 1 <= k <= n.
The number of k's in the first n rows (or in the first A000070(n-1) terms of the sequence) is equal to A000041(n-k), 1 <= k <= n.
The number of terms >= k in the first n rows (or in the first A000070(n-1) terms of the sequence) is equal to A000070(n-k), 1 <= k <= n.
First n rows of triangle (or first A000070(n-1) terms of the sequence) in nonincreasing order give the n-th row of A176206. (End)

			Triangle begins:
1;
2;
3, 1;
4, 2, 1;
5, 3, 2, 1, 1;
6, 4, 3, 2, 2, 1, 1;
7, 5, 4, 3, 3, 2, 2, 1, 1, 1, 1;
8, 6, 5, 4, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1;
9, 7, 6, 5, 5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;
...
For n = 6, by definition the length of row 6 is A000041(6-1) = A000041(5) = 7, so the row 6 of triangle has seven terms. Since every column lists the positive integers A000027 so the row 6 is [6, 4, 3, 2, 2, 1, 1].
Then we have that the divisors of the numbers of the 6th row are:
.
6th row of the triangle ---------->   6 4 3 2 2 1 1
                                      3 2 1 1 1
                                      2 1
                                      1
.
There are seven 1's, four 2's, two 3's, one 4 and one 6.
In total there are 7 + 4 + 2 + 1 + 1 = 15 divisors.
On the other hand the last section of the set of the partitions of 6 can be represented in several ways, five of them as shown below:
._ _ _ _ _ _
|_ _ _      |       6    6                  6                       6
|_ _ _|_    |     3 3    3 3              3   3                     3   3
|_ _    |   |     4 2    4 2            4       2                     4     2
|_ _|_ _|_  |   2 2 2    2 2 2        2   2       2                 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                            1       1
          |_|       1              1                            1   1
.
   Figure 1.  Figure 2.  Figure 3.        Figure 4.                   Figure 5.
.
In every figure there are seven 1's, four 2's, two 3's, one 4 and one 6, as shown also the 6th row of A182703.
In total there are 7 + 4 + 2 + 1 + 1 = A138137(6) = 15 parts in every figure.
Figure 5 is an arrangement that shows the correspondence between divisors and parts since the columns give the divisors of the terms of 6th row of triangle.
Finally we can see that all divisors of all numbers in the 6th row of the triangle are the same positive integers as all parts in the last section of the set of the partitions of 6.
Example edited by _Omar E. Pol_, Aug 10 2021

Paolo Xausa, Table of n, a(n) for n = 1..11732 (rows 1..27 of triangle, flattened).

Row sums give A000070.
Row n has length A000041(n-1).
Every column k gives A000027.
Companion of A176206.
Cf. A000007, A000041, A027750, A028310, A002865, A133735, A135010, A138121, A138137, A182703, A187219, A207378, A221529, A336812, A339278, A340035, A340061, A346741.

A336811[row_]:=Flatten[Table[ConstantArray[row-m,PartitionsP[m]-PartitionsP[m-1]],{m,0,row-1}]];
Array[A336811,10] (* Generates 10 rows *) (* Paolo Xausa, Feb 10 2023 *)

f(n) = numbpart(n-1);
T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (n)); my(s=0); while (k <= f(n-1), s++; n--;); 1+s;}
tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n,k), ", ");); print;);} \\ Michel Marcus, Jan 13 2021

cp's OEIS Frontend

A182703 Triangle read by rows: T(n,k) = number of occurrences of k in the last section of the set of partitions of n.

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A206437 Triangle read by rows: T(j,k) is the k-th part of the j-th region of the set of partitions of n, if 1 <= j <= A000041(n).

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A141285 Largest part of the n-th partition of j in the list of colexicographically ordered partitions of j, if 1 <= n <= A000041(j).

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A138137 First differences of A006128.

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A225620 Indices of partitions in the table of compositions of A228351.

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A187219 Number of partitions of n that do not contain parts less than the smallest part of the partitions of n-1.

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A194446 Number of parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

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