A182703 -id:A182703 - cp's OEIS frontend

A135010 Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of juxtaposed lexicographically ordered partitions of n that do not contain 1 as a part.

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

Offset: 1

Omar E. Pol, Nov 17 2007, Mar 21 2008

This is the original sequence of a large number of sequences connected with the section model of partitions.
Here "the n-th section of the set of partitions of any integer greater than or equal to n" (hence "the last section of the set of partitions of n") is defined to be the set formed by all parts that occur as a result of taking all partitions of n and then removing all parts of the partitions of n-1. For integers greater than 1 the structure of a section has two main areas: the head and tail. The head is formed by the partitions of n that do not contain 1 as a part. The tail is formed by A000041(n-1) partitions of 1. The set of partitions of n contains the sets of partitions of the previous numbers. The section model of partitions has several versions according with the ordering of the partitions or with the representation of the sections. In this sequence we use the ordering of A026791.
The section model of partitions can be interpreted as a table of partitions. See also A138121. - Omar E. Pol, Nov 18 2009
It appears that the versions of the model show an overlapping of sections and subsections of the numbers congruent to k mod m into parts >= m. For example:
First generation (the main table):
Table 1.0: Partitions of integers congruent to 0 mod 1 into parts >= 1.
Second generation:
Table 2.0: Partitions of integers congruent to 0 mod 2 into parts >= 2.
Table 2.1: Partitions of integers congruent to 1 mod 2 into parts >= 2.
Third generation:
Table 3.0: Partitions of integers congruent to 0 mod 3 into parts >= 3.
Table 3.1: Partitions of integers congruent to 1 mod 3 into parts >= 3.
Table 3.2: Partitions of integers congruent to 2 mod 3 into parts >= 3.
And so on.
Conjecture:
Let j and n be integers congruent to k mod m such that 0 <= k < m <= j < n. Let h=(n-j)/m. Consider only all partitions of n into parts >= m. Then remove every partition in which the parts of size m appears a number of times < h. Then remove h parts of size m in every partition. The rest are the partitions of j into parts >= m. (Note that in the section model, h is the number of sections or subsections removed), (Omar E. Pol, Dec 05 2010, Dec 06 2010).
Starting from the first row of triangle, it appears that the total numbers of parts of size k in k successive rows give the sequence A000041 (see A182703). - Omar E. Pol, Feb 22 2012
The last section of n contains A187219(n) regions (see A206437). - Omar E. Pol, Nov 04 2012

			Triangle begins:
  [1];
  [1],[2];
  [1],[1],[3];
  [1],[1],[1],[2,2],[4];
  [1],[1],[1],[1],[1],[2,3],[5];
  [1],[1],[1],[1],[1],[1],[1],[2,2,2],[2,4],[3,3],[6];
  ...
From _Omar E. Pol_, Sep 03 2013: (Start)
Illustration of initial terms (n = 1..6). The table shows the six sections of the set of partitions of 6 in three ways. Note that before the dissection, the set of partitions was in the ordering mentioned in A026791. More generally, the six sections of the set of partitions of 6 also can be interpreted as the first six sections of the set of partitions of any integer >= 6.
---------------------------------------------------------
n  j          Diagram          Parts           Parts
---------------------------------------------------------
.                   _
1  1               |_|                1;              1;
.                 _
2  1             | |_               1,              1,
2  2             |_ _|              2;                2;
.               _
3  1           | |                1,              1,
3  2           | |_ _             1,                1,
3  3           |_ _ _|            3;                  3;
.             _
4  1         | |                1,              1,
4  2         | |                1,                1,
4  3         | |_ _ _           1,                  1,
4  4         |   |_ _|          2,2,                2,2,
4  5         |_ _ _ _|          4;                    4;
.           _
5  1       | |                1,              1,
5  2       | |                1,                1,
5  3       | |                1,                  1,
5  4       | |                1,                  1,
5  5       | |_ _ _ _         1,                    1,
5  6       |   |_ _ _|        2,3,                  2,3,
5  7       |_ _ _ _ _|        5;                      5;
.         _
6  1     | |                1,              1,
6  2     | |                1,                1,
6  3     | |                1,                  1,
6  4     | |                1,                  1,
6  5     | |                1,                    1,
6  6     | |                1,                    1,
6  7     | |_ _ _ _ _       1,                      1,
6  8     |   |   |_ _|      2,2,2,                2,2,2,
6  9     |   |_ _ _ _|      2,4,                    2,4,
6  10    |     |_ _ _|      3,3,                    3,3,
6  11    |_ _ _ _ _ _|      6;                        6;
...
(End)

Alois P. Heinz, Rows n = 1..23, flattened
Omar E. Pol, Illustration of the section model of partitions
Omar E. Pol, Illustration of the section model of partitions (2D view)
Omar E. Pol, Illustration of the section model of partitions (3D view)
Robert Price, Mathematica program to generate "Illustration of the section model of partitions"
Robert Price, Mathematica program to generate "Illustration of the section model of partitions (2D view)"

Row n has length A138137(n).
Row sums give A138879.
Right border gives A000027.
Cf. A000041, A026791, A138121, A141285, A182703, A187219, A193870, A194446, A206437, A207031, A207383, A207379, A211009.

with(combinat):
T:= proc(m) local b, ll;
      b:= proc(n, i, l)
            if n=0 then ll:=ll, l[]
          else seq(b(n-j, j, [l[], j]), j=i..n)
            fi
          end;
      ll:= NULL; b(m, 2, []); [1$numbpart(m-1)][], ll
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Feb 19 2012

less[run1_, run2_] := (lg1 = run1 // Length; lg2 = run2 // Length; lg = Max[lg1, lg2]; r1 = If[lg1 == lg, run1, PadRight[run1, lg, 0]]; r2 = If[lg2 == lg, run2, PadRight[run2, lg, 0]]; Order[r1, r2] != -1); row[n_] := Join[ Array[1 &, {PartitionsP[n - 1]}], Sort[ Reverse /@ Select[ IntegerPartitions[n], FreeQ[#, 1] &], less] ] // Flatten; Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Jan 14 2013 *)
Table[Reverse@ConstantArray[{1}, PartitionsP[n - 1]]~Join~
DeleteCases[Sort@PadRight[Reverse/@Cases[IntegerPartitions[n], x_ /; Last[x] != 1]], x_ /; x == 0, 2], {n, 1, 9}] // Flatten (* Robert Price, May 12 2020 *)

A138121 Triangle read by rows in which row n lists the partitions of n that do not contain 1 as a part in juxtaposed reverse-lexicographical order followed by A000041(n-1) 1's.

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

Offset: 1

Omar E. Pol, Mar 21 2008

Mirror of triangle A135010.

			Triangle begins:
[1];
[2],[1];
[3],[1],[1];
[4],[2,2],[1],[1],[1];
[5],[3,2],[1],[1],[1],[1],[1];
[6],[3,3],[4,2],[2,2,2],[1],[1],[1],[1],[1],[1],[1];
[7],[4,3],[5,2],[3,2,2],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1];
...
The illustration of the three views of the section model of partitions (version "tree" with seven sections) shows the connection between several sequences.
---------------------------------------------------------
Partitions                A194805            Table 1.0
.  of 7       p(n)        A194551             A135010
---------------------------------------------------------
7              15                    7     7 . . . . . .
4+3                                4       4 . . . 3 . .
5+2                              5         5 . . . . 2 .
3+2+2                          3           3 . . 2 . 2 .
6+1            11    6       1             6 . . . . . 1
3+3+1                  3     1             3 . . 3 . . 1
4+2+1                    4   1             4 . . . 2 . 1
2+2+2+1                    2 1             2 . 2 . 2 . 1
5+1+1           7            1   5         5 . . . . 1 1
3+2+1+1                      1 3           3 . . 2 . 1 1
4+1+1+1         5        4   1             4 . . . 1 1 1
2+2+1+1+1                  2 1             2 . 2 . 1 1 1
3+1+1+1+1       3            1 3           3 . . 1 1 1 1
2+1+1+1+1+1     2          2 1             2 . 1 1 1 1 1
1+1+1+1+1+1+1   1            1             1 1 1 1 1 1 1
.               1                         ---------------
.               *<------- A000041 -------> 1 1 2 3 5 7 11
.                         A182712 ------->   1 0 2 1 4 3
.                         A182713 ------->     1 0 1 2 2
.                         A182714 ------->       1 0 1 1
.                                                  1 0 1
.                         A141285           A182703  1 0
.                    A182730   A182731                 1
---------------------------------------------------------
.                              A138137 --> 1 2 3 6 9 15..
---------------------------------------------------------
.       A182746 <--- 4 . 2 1 0 1 2 . 4 ---> A182747
---------------------------------------------------------
.
.       A182732 <--- 6 3 4 2 1 3 5 4 7 ---> A182733
.                    . . . . 1 . . . .
.                    . . . 2 1 . . . .
.                    . 3 . . 1 2 . . .
.      Table 2.0     . . 2 2 1 . . 3 .     Table 2.1
.                    . . . . 1 2 2 . .
.                            1 . . . .
.
.  A182982  A182742       A194803       A182983  A182743
.  A182992  A182994       A194804       A182993  A182995
---------------------------------------------------------
.
From _Omar E. Pol_, Sep 03 2013: (Start)
Illustration of initial terms (n = 1..6). The table shows the six sections of the set of partitions of 6. Note that before the dissection the set of partitions was in the ordering mentioned in A026792. More generally, the six sections of the set of partitions of 6 also can be interpreted as the first six sections of the set of partitions of any integer >= 6.
Illustration of initial terms:
---------------------------------------
n  j     Diagram          Parts
---------------------------------------
.         _
1  1     |_|              1;
.         _ _
2  1     |_  |            2,
2  2       |_|            .  1;
.         _ _ _
3  1     |_ _  |          3,
3  2         | |          .  1,
3  3         |_|          .  .  1;
.         _ _ _ _
4  1     |_ _    |        4,
4  2     |_ _|_  |        2, 2,
4  3           | |        .  1,
4  4           | |        .  .  1,
4  5           |_|        .  .  .  1;
.         _ _ _ _ _
5  1     |_ _ _    |      5,
5  2     |_ _ _|_  |      3, 2,
5  3             | |      .  1,
5  4             | |      .  .  1,
5  5             | |      .  .  1,
5  6             | |      .  .  .  1,
5  7             |_|      .  .  .  .  1;
.         _ _ _ _ _ _
6  1     |_ _ _      |    6,
6  2     |_ _ _|_    |    3, 3,
6  3     |_ _    |   |    4, 2,
6  4     |_ _|_ _|_  |    2, 2, 2,
6  5               | |    .  1,
6  6               | |    .  .  1,
6  7               | |    .  .  1,
6  8               | |    .  .  .  1,
6  9               | |    .  .  .  1,
6  10              | |    .  .  .  .  1,
6  11              |_|    .  .  .  .  .  1;
...
(End)

Robert Price, Table of n, a(n) for n = 1..16851, 25 rows
Omar E. Pol, A section model of partitions (2D and 3D) [From _Omar E. Pol_, Sep 07 2008]
Robert Price, Mathematica Program to Generate Diagram

Row n has length A138137(n).
Rows sums give A138879.
Cf. A000041, A135010, A138879, A138880, A141285, A182703, A194812, A206437, A211009.

less[run1_, run2_] := (lg1 = run1 // Length; lg2 = run2 // Length; lg = Max[lg1, lg2]; r1 = If[lg1 == lg, run1, PadRight[run1, lg, 0]]; r2 = If[lg2 == lg, run2, PadRight[run2, lg, 0]]; Order[r1, r2] != -1); row[n_] := Join[Array[1 &, {PartitionsP[n - 1]}], Sort[Reverse /@ Select[IntegerPartitions[n], FreeQ[#, 1] &], less]] // Flatten // Reverse; Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Jan 15 2013 *)
Table[Reverse/@Reverse@DeleteCases[Sort@PadRight[Reverse/@Cases[IntegerPartitions[n], x_ /; Last[x]!=1]], x_ /; x==0, 2]~Join~ConstantArray[{1}, PartitionsP[n - 1]], {n, 1, 9}]  // Flatten (* Robert Price, May 11 2020 *)

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

A066633 Triangle T(n,k), n >= 1, 1 <= k <= n, giving number of k's in all partitions of n.

Original entry on oeis.org

1, 2, 1, 4, 1, 1, 7, 3, 1, 1, 12, 4, 2, 1, 1, 19, 8, 4, 2, 1, 1, 30, 11, 6, 3, 2, 1, 1, 45, 19, 9, 6, 3, 2, 1, 1, 67, 26, 15, 8, 5, 3, 2, 1, 1, 97, 41, 21, 13, 8, 5, 3, 2, 1, 1, 139, 56, 31, 18, 12, 7, 5, 3, 2, 1, 1, 195, 83, 45, 28, 17, 12, 7, 5, 3, 2, 1, 1, 272, 112, 63, 38, 25, 16, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Naohiro Nomoto, Jan 09 2002

It appears that row n lists the first differences of the row n of triangle A181187 together with 1 (as the final term of the row n). - Omar E. Pol, Feb 26 2012
It appears that reversed rows converge to A000041. - Omar E. Pol, Mar 11 2012
Proof: For a partition of n with k>floor(n/2+1), k can only occur as the largest part; the other parts sum to n-k, so that T(n,n-k)=A000041(k). - George Beck, Jun 30 2019
T(n,k) is also the total number k's that are divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. - Omar E. Pol, Feb 05 2021

			For n = 3, k = 1; 3 = 2+1 = 1+1+1. T(3,1) = (zero 1's) + (one 1's) + (three 1's), so T(3,1) = 4.
Triangle begins:
    1;
    2,   1;
    4,   1,  1;
    7,   3,  1,  1;
   12,   4,  2,  1,  1;
   19,   8,  4,  2,  1,  1;
   30,  11,  6,  3,  2,  1,  1;
   45,  19,  9,  6,  3,  2,  1, 1;
   67,  26, 15,  8,  5,  3,  2, 1, 1;
   97,  41, 21, 13,  8,  5,  3, 2, 1, 1;
  139,  56, 31, 18, 12,  7,  5, 3, 2, 1, 1;
  195,  83, 45, 28, 17, 12,  7, 5, 3, 2, 1, 1;
  272, 112, 63, 38, 25, 16, 11, 7, 5, 3, 2, 1, 1;
  ...

D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.2.1.5, Problem 73(b), pp. 415, 761. - N. J. A. Sloane, Dec 30 2018

Alois P. Heinz, Rows n = 1..141, flattened
Manosij Ghosh Dastidar and Sourav Sen Gupta, Generalization of a few results in Integer Partitions, arXiv preprint arXiv:1111.0094 [cs.DM], 2011.
Joseph Vandehey, Digital problems in the theory of partitions, Integers (2024) Vol. 24A, Art. No. A18. See p. 3.
Eric Weisstein's World of Mathematics, Elder's Theorem

Row sums give positive terms of A006128.

Columns (1-10): A000070, A024786-A024794.

b:= proc(n, i) option remember;
      `if`(n=0 or i=1, 1+n*x, b(n, i-1)+
      `if`(i>n, 0, (g->g+coeff(g, x, 0)*x^i)(b(n-i, i))))
    end:
T:= n-> (p->seq(coeff(p, x, i), i=1..n))(b(n, n)):
seq(T(n), n=1..14);  # Alois P. Heinz, Mar 21 2012

Table[Count[Flatten[IntegerPartitions[n]],k],
{n,1,20},{k,1,n}]
TableForm[% ] (* as a triangle *)
Flatten[%%]   (* as a sequence *)
(* Clark Kimberling, Mar 03 2010 *)
T[n_, n_] = 1; T[n_, k_] /; k, ] = 0; Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 21 2015, after Omar E. Pol *)

from math import isqrt, comb
from sympy import partition
def A066633(n):
    a = (m:=isqrt(k:=n<<1))+(k>m*(m+1))
    b = n-comb(a,2)
    return sum(partition(j) for j in range(a%b,a,b)) # Chai Wah Wu, Nov 13 2024

G.f. for the number of k's in all partitions of n is x^k/(1-x^k)* Product_{m>=1} 1/(1-x^m). - Vladeta Jovovic, Jan 15 2002

T(n, k) = Sum_{j

Equals triangle A027293 * A051731 as infinite lower triangular matrices. - Gary W. Adamson Mar 21 2011

It appears that T(n+k,k) = T(n,k) + A000041(n). - Omar E. Pol, Feb 04 2012. This was proved in the Dastidar-Gupta paper in Lemma 1. - George Beck, Jun 26 2019

It appears that T(n,k) = A206563(n,k) - A206563(n,k+2). - Omar E. Pol, Feb 26 2012

T(n,k) = Sum_{j=1..n} A182703(j,k). - Omar E. Pol, May 02 2012

More terms from Vladeta Jovovic, Jan 11 2002

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

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

A221529 Triangle read by rows: T(n,k) = A000203(k)*A000041(n-k), 1 <= k <= n.

Original entry on oeis.org

1, 1, 3, 2, 3, 4, 3, 6, 4, 7, 5, 9, 8, 7, 6, 7, 15, 12, 14, 6, 12, 11, 21, 20, 21, 12, 12, 8, 15, 33, 28, 35, 18, 24, 8, 15, 22, 45, 44, 49, 30, 36, 16, 15, 13, 30, 66, 60, 77, 42, 60, 24, 30, 13, 18, 42, 90, 88, 105, 66, 84, 40, 45, 26, 18, 12, 56, 126, 120, 154, 90, 132, 56, 75, 39, 36, 12, 28

Offset: 1

Omar E. Pol, Jan 20 2013

Since A000203(k) has a symmetric representation, both T(n,k) and the partial sums of row n can be represented by symmetric polycubes. For more information see A237593 and A237270. For another version see A245099. - Omar E. Pol, Jul 15 2014
From Omar E. Pol, Jul 10 2021: (Start)
The above comment refers to a symmetric tower whose terraces are the symmetric representation of sigma(i), for i = 1..n, starting from the top. The levels of these terraces are the partition numbers A000041(h-1), for h = 1 to n, starting from the base of the tower, where n is the length of the largest side of the base.
The base of the tower is the symmetric representation of A024916(n).
The height of the tower is equal to A000041(n-1).
The surface area of the tower is equal to A345023(n).
The volume (or the number of cubes) of the tower equals A066186(n).
The volume represents the n-th term of the convolution of A000203 and A000041, that is A066186(n).
Note that the terraces that are the symmetric representation of sigma(n) and the terraces that are the symmetric representation of sigma(n-1) both are unified in level 1 of the structure. That is because the first two partition numbers A000041 are [1, 1].
The tower is an object of the family of the stepped pyramid described in A245092.
T(n,k) can be represented with a set of A237271(k) right prisms of height A000041(n-k) since T(n,k) is the total number of cubes that are exactly below the parts of the symmetric representation of sigma(k) in the tower.
T(n,k) is also the sum of all divisors of all k's that are in the first n rows of triangle A336811, or in other words, in the first A000070(n-1) terms of the sequence A336811. Hence T(n,k) is also the sum of all divisors of all k's in the n-th row of triangle A176206.
The mentioned property is due to the correspondence between divisors and parts explained in A338156: all divisors of the first A000070(n-1) terms of A336811 are also all parts of all partitions of n.
Therefore the set of all partitions of n >= 1 has an associated tower.
The partial column sums of A340583 give this triangle showing the growth of the structure of the tower.
Note that the convolution of A000203 with any integer sequence S can be represented with a symmetric tower or structure of the same family where its terraces are the symmetric representation of sigma starting from the top and the heights of the terraces starting from the base are the terms of the sequence S. (End)

			Triangle begins:
------------------------------------------------------
    n| k    1   2   3   4   5   6   7   8   9  10
------------------------------------------------------
    1|      1;
    2|      1,  3;
    3|      2,  3,  4;
    4|      3,  6,  4,  7;
    5|      5,  9,  8,  7,  6;
    6|      7, 15, 12, 14,  6, 12;
    7|     11, 21, 20, 21, 12, 12,  8;
    8|     15, 33, 28, 35, 18, 24,  8, 15;
    9|     22, 45, 44, 49, 30, 36, 16, 15, 13;
   10|     30, 66, 60, 77, 42, 60, 24, 30, 13, 18;
...
The sum of row 10 is [30 + 66 + 60 + 77 + 42 + 60 + 24 + 30 + 13 + 18] = A066186(10) = 420.
.
For n = 10 the calculation of the row 10 is as follows:
    k    A000203         T(10,k)
    1       1   *  30   =   30
    2       3   *  22   =   66
    3       4   *  15   =   60
    4       7   *  11   =   77
    5       6   *   7   =   42
    6      12   *   5   =   60
    7       8   *   3   =   24
    8      15   *   2   =   30
    9      13   *   1   =   13
   10      18   *   1   =   18
                 A000041
.
From _Omar E. Pol_, Jul 13 2021: (Start)
For n = 10 we can see below three views of two associated polycubes called here "prism of partitions" and "tower". Both objects contain the same number of cubes (that property is valid for n >= 1).
        _ _ _ _ _ _ _ _ _ _
  42   |_ _ _ _ _          |
       |_ _ _ _ _|_        |
       |_ _ _ _ _ _|_      |
       |_ _ _ _      |     |
       |_ _ _ _|_ _ _|_    |
       |_ _ _ _        |   |
       |_ _ _ _|_      |   |
       |_ _ _ _ _|_    |   |
       |_ _ _      |   |   |
       |_ _ _|_    |   |   |
       |_ _    |   |   |   |
       |_ _|_ _|_ _|_ _|_  |                             _
  30   |_ _ _ _ _        | |                            | | 30
       |_ _ _ _ _|_      | |                            | |
       |_ _ _      |     | |                            | |
       |_ _ _|_ _ _|_    | |                            | |
       |_ _ _ _      |   | |                            | |
       |_ _ _ _|_    |   | |                            | |
       |_ _ _    |   |   | |                            | |
       |_ _ _|_ _|_ _|_  | |                           _|_|
  22   |_ _ _ _        | | |                          |   |  22
       |_ _ _ _|_      | | |                          |   |
       |_ _ _ _ _|_    | | |                          |   |
       |_ _ _      |   | | |                          |   |
       |_ _ _|_    |   | | |                          |   |
       |_ _    |   |   | | |                          |   |
       |_ _|_ _|_ _|_  | | |                         _|_ _|
  15   |_ _ _ _      | | | |                        | |   |  15
       |_ _ _ _|_    | | | |                        | |   |
       |_ _ _    |   | | | |                        | |   |
       |_ _ _|_ _|_  | | | |                       _|_|_ _|
  11   |_ _ _      | | | | |                      | |     |  11
       |_ _ _|_    | | | | |                      | |     |
       |_ _    |   | | | | |                      | |     |
       |_ _|_ _|_  | | | | |                     _| |_ _ _|
   7   |_ _ _    | | | | | |                    |   |     |   7
       |_ _ _|_  | | | | | |                   _|_ _|_ _ _|
   5   |_ _    | | | | | | |                  | | |       |   5
       |_ _|_  | | | | | | |                 _| | |_ _ _ _|
   3   |_ _  | | | | | | | |               _|_ _|_|_ _ _ _|   3
   2   |_  | | | | | | | | |           _ _|_ _|_|_ _ _ _ _|   2
   1   |_|_|_|_|_|_|_|_|_|_|          |_ _|_|_|_ _ _ _ _ _|   1
.
             Figure 1.                       Figure 2.
         Front view of the                 Lateral view
        prism of partitions.               of the tower.
.
.                                      _ _ _ _ _ _ _ _ _ _
                                      |   | | | | | | | |_|   1
                                      |   | | | | | |_|_ _|   2
                                      |   | | | |_|_  |_ _|   3
                                      |   | |_|_    |_ _ _|   4
                                      |   |_ _  |_  |_ _ _|   5
                                      |_ _    |_  |_ _ _ _|   6
                                          |_    | |_ _ _ _|   7
                                            |_  |_ _ _ _ _|   8
                                              |           |   9
                                              |_ _ _ _ _ _|  10
.
                                             Figure 3.
                                             Top view
                                           of the tower.
.
Figure 1 is a two-dimensional diagram of the partitions of 10 in colexicographic order (cf. A026792, A211992). The area of the diagram is 10*42 = A066186(10) = 420. Note that the diagram can be interpreted also as the front view of a right prism whose volume is 1*10*42 = 420 equaling the volume and the number of cubes of the tower that appears in the figures 2 and 3.
Note that the shape and the area of the lateral view of the tower are the same as the shape and the area where the 1's are located in the diagram of partitions. In this case the mentioned area equals A000070(10-1) = 97.
The connection between these two associated objects is a representation of the correspondence divisor/part described in A338156. See also A336812.
The sum of the volumes of both objects equals A220909.
For the connection with the table of A338156 see also A340035. (End)

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened)
T. J. Osler, A. Hassen and T. R. Chandrupatia, Surprising connections between partitions and divisors, The College Mathematics Journal, Vol. 38. No. 4, Sep. 2007, 278-287 (see p. 287).
Omar E. Pol, Illustration of the prism, the tower and the 10th row of the triangle

Row sums give A066186.
Column 1 is A000041.
Leading diagonals 1-5: A000203, A000203, A074400, A272027, A274535.
Companion of A221530.
Cf. A000070, A000203, A026792, A027293, A135010, A138137, A176206, A182703, A220909, A211992, A221649, A236104, A237270, A237271, A237593, A245092, A245093, A245095, A245099, A262626, A336811, A336812, A338156, A339278, A340035, A340583, A340584, A345023, A346741.

nrows=12; Table[Table[DivisorSigma[1,k]PartitionsP[n-k],{k,n}],{n,nrows}] // Flatten (* Paolo Xausa, Jun 17 2022 *)

T(n,k)=sigma(k)*numbpart(n-k) \\ Charles R Greathouse IV, Feb 19 2013

T(n,k) = sigma(k)*p(n-k) = A000203(k)*A027293(n,k).

T(n,k) = A245093(n,k)*A027293(n,k).

A338156 Irregular triangle read by rows in which row n lists n blocks, where the m-th block consists of A000041(m-1) copies of the divisors of (n - m + 1), with 1 <= m <= n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Oct 14 2020

In other words: in row n replace every term of n-th row of A176206 with its divisors.
The terms in row n are also all parts of all partitions of n.
As in A336812 here we introduce a new type of table which shows the correspondence between divisors and partitions. More precisely the table shows the correspondence between all divisors of all terms of the n-th row of A176206 and all parts of all partitions of n, with n >= 1. Both the mentionded divisors and the mentioned parts are the same numbers (see Example section). That is because all divisors of the first A000070(n-1) terms of A336811 are also all parts of all partitions of n.
For an equivalent table for all parts of the last section of the set of partitions of n see the subsequence A336812. The section is the smallest substructure of the set of partitions in which appears the correspondence divisor/part.
From Omar E. Pol, Aug 01 2021: (Start)
The terms of row n appears in the triangle A346741 ordered in accordance with the successive sections of the set of partitions of n.
The terms of row n in nonincreasing order give the n-th row of A302246.
The terms of row n in nondecreasing order give the n-th row of A302247.
For the connection with the tower described in A221529 see also A340035. (End)

			Triangle begins:
  [1];
  [1,2],   [1];
  [1,3],   [1,2],   [1],   [1];
  [1,2,4], [1,3],   [1,2], [1,2], [1],   [1],   [1];
  [1,5],   [1,2,4], [1,3], [1,3], [1,2], [1,2], [1,2], [1], [1], [1], [1], [1];
  ...
For n = 5 the 5th row of A176206 is [5, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1, 1] so replacing every term with its divisors we have the 5th row of this triangle.
Also, if the sequence is written as an irregular tetrahedron so the first six slices are:
  [1],
  -------
  [1, 2],
  [1],
  -------
  [1, 3],
  [1, 2],
  [1],
  [1];
  ----------
  [1, 2, 4],
  [1, 3],
  [1, 2],
  [1, 2],
  [1],
  [1],
  [1];
  ----------
  [1, 5],
  [1, 2, 4],
  [1, 3],
  [1, 3],
  [1, 2],
  [1, 2],
  [1, 2],
  [1],
  [1],
  [1],
  [1],
  [1];
.
The above slices appear in the lower zone of the following table which shows the correspondence between the mentioned divisors and all parts of all partitions of the positive integers.
The table is infinite. It is formed by three zones as follows:
The upper zone shows the partitions of every positive integer in colexicographic order (cf. A026792, A211992).
The lower zone shows the same numbers but arranged as divisors in accordance with the slices of the tetrahedron mentioned above.
Finally the middle zone shows the connection between the upper zone and the lower zone.
For every positive integer the numbers in the upper zone are the same numbers as in the lower zone.
.
|---|---------|-----|-------|---------|------------|---------------|
| 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 | A066633 |  1  |  2 1  |  4 1 1  |  7  3 1 1  | 12  4  2 1 1  |
| I |         |  *  |  * *  |  * * *  |  *  * * *  |  *  *  * * *  |
| N | A002260 |  1  |  1 2  |  1 2 3  |  1  2 3 4  |  1  2  3 4 5  |
| K |         |  =  |  = =  |  = = =  |  =  = = =  |  =  =  = = =  |
|   | A138785 |  1  |  2 2  |  4 2 3  |  7  6 3 4  | 12  8  6 4 5  |
|   |         |  |  |  |\|  |  |\|\|  |  |\ |\|\|  |  |\ |\ |\|\|  |
|   | A206561 |  1  |  4 2  |  9 5 3  | 20 13 7 4  | 35 23 15 9 5  |
|---|---------|-----|-------|---------|------------|---------------|
.
|---|---------|-----|-------|---------|------------|---------------|
|   | A027750 |  1  |  1 2  |  1   3  |  1  2   4  |  1         5  |
|   |---------|-----|-------|---------|------------|---------------|
|   | A027750 |     |  1    |  1 2    |  1    3    |  1  2    4    |
|   |---------|-----|-------|---------|------------|---------------|
| D | A027750 |     |       |  1      |  1  2      |  1     3      |
| I | A027750 |     |       |  1      |  1  2      |  1     3      |
| V |---------|-----|-------|---------|------------|---------------|
| I | A027750 |     |       |         |  1         |  1  2         |
| S | A027750 |     |       |         |  1         |  1  2         |
| O | A027750 |     |       |         |  1         |  1  2         |
| R |---------|-----|-------|---------|------------|---------------|
| S | A027750 |     |       |         |            |  1            |
|   | A027750 |     |       |         |            |  1            |
|   | A027750 |     |       |         |            |  1            |
|   | A027750 |     |       |         |            |  1            |
|   | A027750 |     |       |         |            |  1            |
|---|---------|-----|-------|---------|------------|---------------|
.
Note that every row in the lower zone lists A027750.
Also the lower zone for every positive integer can be constructed using the first n terms of the partition numbers. For example: for n = 5 we consider the first 5 terms of A000041 (that is [1, 1, 2, 3, 5]) then the 5th slice is formed by a block with the divisors of 5, one block with the divisors of 4, two blocks with the divisors of 3, three blocks with the divisors of 2, and five blocks with the divisors of 1.
Note that the lower zone is also in accordance with the tower (a polycube) described in A221529 in which its terraces are the symmetric representation of sigma starting from the top (cf. A237593) and the heights of the mentioned terraces are the partition numbers A000041 starting from the base.
The tower has the same volume (also the same number of cubes) equal to A066186(n) as a prism of partitions of size 1*n*A000041(n).
The above table shows the correspondence between the prism of partitions and its associated tower since the number of parts in all partitions of n is equal to A006128(n) equaling the number of divisors in the n-th slice of the lower table and equaling the same the number of terms in the n-th row of triangle. Also the sum of all parts of all partitions of n is equal to A066186(n) equaling the sum of all divisors in the n-th slice of the lower table and equaling the sum of the n-th row of triangle.

Paolo Xausa, Table of n, a(n) for n = 1..13185 (rows 1..19 of triangle, flattened)

Nonzero terms of A340031.
Row n has length A006128(n).
The sum of row n is A066186(n).
The product of row n is A007870(n).
Row n lists the first n rows of A336812 (a subsequence).
The number of parts k in row n is A066633(n,k).
The sum of all parts k in row n is A138785(n,k).
The number of parts >= k in row n is A181187(n,k).
The sum of all parts >= k in row n is A206561(n,k).
The number of parts <= k in row n is A210947(n,k).
The sum of all parts <= k in row n is A210948(n,k).
Cf. A000070, A000041, A002260, A026792, A027750, A058399, A127093, A135010, A138121, A176206, A182703, A206437, A207031, A207383, A211992, A221529, A221530, A221531, A245095, A221649, A221650, A237593, A302246, A302247, A336811, A337209, A339106, A339258, A339278, A339304, A340035, A340061, A346741.

A338156[rowmax_]:=Table[Flatten[Table[ConstantArray[Divisors[n-m],PartitionsP[m]],{m,0,n-1}]],{n,rowmax}];
A338156[10] (* Generates 10 rows *) (* Paolo Xausa, Jan 12 2023 *)

A338156(rowmax)=vector(rowmax,n,concat(vector(n,m,concat(vector(numbpart(m-1),i,divisors(n-m+1))))));
A338156(10) \\ Generates 10 rows - Paolo Xausa, Feb 17 2023

A207031 Triangle read by rows: T(n,k) = sum of all parts of the k-th column of the last section of the set of partitions of n.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 6, 3, 1, 1, 8, 3, 2, 1, 1, 15, 8, 4, 2, 1, 1, 19, 8, 5, 3, 2, 1, 1, 32, 17, 9, 6, 3, 2, 1, 1, 42, 20, 13, 7, 5, 3, 2, 1, 1, 64, 34, 19, 13, 8, 5, 3, 2, 1, 1, 83, 41, 26, 16, 11, 7, 5, 3, 2, 1, 1, 124, 68, 41, 27, 17, 12, 7, 5, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Feb 14 2012

Also T(n,k) is the number of parts >= k in the last section of the set of partitions of n. Therefore T(n,1) = A138137(n), the total number of parts in the last section of the set of partitions of n. For calculation of the number of odd/even parts, etc, follow the same rules from A206563.
More generally, let m and n be two positive integers such that m <= n. It appears that any set formed by m connected sections, or m disconnected sections, or a mixture of both, has the same properties described in the entry A206563.
It appears that reversed rows converge to A000041.
It appears that the first differences of row n together with 1 give the row n of triangle A182703 (see example). - Omar E. Pol, Feb 26 2012

			Illustration of initial terms. First six rows of triangle as sums of columns from the last sections of the first six natural numbers (or as sums of columns from the 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
. --- --- ------- --------- ----------- --------------
A: 1, 2,1, 3,1,1,  6,3,1,1,  8,3,2,1,1,  15,8,4,2,1,1
.  |  |/|  |/|/|   |/|/|/|   |/|/|/|/|    |/|/|/|/|/|
B: 1, 1,1, 2,0,1,  3,2,0,1,  5,1,1,0,1,   7,4,2,1,0,1
.
A := initial terms of this triangle.
B := initial terms of triangle A182703.
.
Triangle begins:
1;
2,    1;
3,    1,  1;
6,    3,  1,  1;
8,    3,  2,  1,  1;
15,   8,  4,  2,  1,  1;
19,   8,  5,  3,  2,  1,  1;
32,  17,  9,  6,  3,  2,  1,  1;
42,  20, 13,  7,  5,  3,  2,  1,  1;
64,  34, 19, 13,  8,  5,  3,  2,  1,  1;
83,  41, 26, 16, 11,  7,  5,  3,  2,  1,  1;
124, 68, 41, 27, 17, 12,  7,  5,  3,  2,  1,  1;

Columns (1-2): A138137, A138135.
Row sums give A138879.
Cf. A000041, A002865, A006128, A135010, A138121, A181187, A182703, A206562, A206563, A207032, A207379, A208476, A210955, A210956.

From Omar E. Pol, Dec 07 2019: (Start)
From the formula in A138135 (year 2008) we have that:
A000041(n-1) = A138137(n) - A138135(n) = T(n,1) - T(n,2);
Hence A000041(n) = T(n+1,1) - T(n+1,2), n >= 0;
Also  A000041(n) = A002865(n) + T(n,1) - T(n,2). (End)

More terms from Alois P. Heinz, Feb 17 2012

Showing 1-10 of 99 results. Next

cp's OEIS Frontend

A135010 Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of juxtaposed lexicographically ordered partitions of n that do not contain 1 as a part.

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A138121 Triangle read by rows in which row n lists the partitions of n that do not contain 1 as a part in juxtaposed reverse-lexicographical order followed by A000041(n-1) 1's.

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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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A066633 Triangle T(n,k), n >= 1, 1 <= k <= n, giving number of k's in all partitions of n.

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A138137 First differences of A006128.

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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.

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