A036042 -id:A036042 - cp's OEIS frontend

A036041 Number of prime divisors, counted with multiplicity, of prime signature A025487(n); equals size of associated partition.

0, 1, 2, 2, 3, 3, 4, 4, 3, 5, 4, 5, 4, 6, 5, 6, 5, 7, 6, 5, 7, 4, 6, 6, 8, 7, 6, 8, 5, 7, 7, 9, 8, 7, 9, 6, 8, 6, 8, 10, 7, 9, 6, 8, 8, 10, 7, 9, 7, 9, 11, 8, 10, 5, 7, 9, 9, 11, 8, 10, 8, 10, 12, 9, 11, 6, 8, 10, 8, 10, 12, 7, 9, 9, 11, 9, 8, 11, 10, 13, 10, 12, 7, 9, 11, 9, 11, 13, 8, 10, 10, 12

Offset: 1

Alford Arnold

			a(3) = 2 since A025487(3) = 4 = 2*2; a(5) = 3 since A025487(5) = 8 = 2*2*2; ...

Álvar Ibeas, Table of n, a(n) for n = 1..10000

Cf. A025487, A000041, A035796, A036042, A001222.

a(n) = A001222(A025487(n)) = A001222(A181822(n)).

More terms from Henry Bottomley, Apr 30 2001
Edited to accommodate change in A025487's offset by Matthew Vandermast, Nov 08 2008
Definition corrected by Álvar Ibeas, Nov 01 2014

A211999 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 14 2012

The sequence lists the partitions of all positive integers. Each row of the irregular array is a partition of j.
At stage 1, we start with 1.
At stage j > 1, we write the partitions of j using the following rules:
First we copy the last A000041(j-1) rows of the array in descending order, as a mirror image, starting with the row that contains the part of size j-1. At the end of each new row, we added a part of size 1.
Second, we write the partitions of j that do not contain 1 as a part, in reverse-lexicographic order, such that the last row (or partition of j) is j.
Note that the table can be partially folded. In this case we have a three-dimensional structure in which each column contains parts of the same size (see example). Also the table can be completely folded, therefore stacked parts have the same size.

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

Omar E. Pol, Illustration of initial terms

Rows sums give A036042, n>=1.
Other versions are A211983, A211984, A211989. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A333486 Length of the n-th reversed integer partition in graded reverse-lexicographic order. Partition lengths of A228531.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 23 2020

			Triangle begins:
  0
  1
  1 2
  1 2 3
  1 2 2 3 4
  1 2 2 3 3 4 5
  1 2 2 3 2 3 3 4 4 5 6
  1 2 2 3 2 3 3 4 3 4 4 5 5 6 7
  1 2 2 2 3 3 4 2 3 3 4 3 4 4 5 4 5 5 6 6 7 8

OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The generalization to compositions is A000120.
Row sums are A006128.
The same partition has sum A036042.
The length-sensitive version (sum/length/revlex) is A036043.
The colexicographic version (sum/colex) is A049085.
The same partition has minimum A182715.
The lexicographic version (sum/lex) is A193173.
The tetrangle of these partitions is A228531.
The version for non-reversed partitions is A238966.
The same partition has Heinz number A334436.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
Partitions in lexicographic order (sum/lex) are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Partitions in opposite Abramowitz-Stegun order (sum/length/revlex) are A334439.
Cf. A026792, A049085, A080576, A080577, A103921, A112798, A115623, A129129, A331581, A334302, A334435, A334442.

revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
Table[Length/@Sort[Reverse/@IntegerPartitions[n],revlexsort],{n,0,8}]

A019575 Place n distinguishable balls in n boxes (in n^n ways); let T(n,k) = number of ways that the maximum in any box is k, for 1 <= k <= n; sequence gives triangle of numbers T(n,k).

Original entry on oeis.org

1, 2, 2, 6, 18, 3, 24, 180, 48, 4, 120, 2100, 800, 100, 5, 720, 28800, 14700, 2250, 180, 6, 5040, 458640, 301350, 52920, 5292, 294, 7, 40320, 8361360, 6867840, 1342600, 153664, 10976, 448, 8, 362880, 172141200, 172872000, 36991080, 4644864, 387072, 20736, 648, 9

Offset: 1

Lee Corbin (lcorbin(AT)tsoft.com)

T(n,k) is the number of endofunctions on [n] such that the maximal cardinality of the nonempty preimages equals k. - Alois P. Heinz, Jul 31 2014

			Triangle begins:
       1;
       2,         2;
       6,        18,         3;
      24,       180,        48,        4;
     120,      2100,       800,      100,       5;
     720,     28800,     14700,     2250,     180,      6;
    5040,    458640,    301350,    52920,    5292,    294,     7;
   40320,   8361360,   6867840,  1342600,  153664,  10976,   448,   8;
  362880, 172141200, 172872000, 36991080, 4644864, 387072, 20736, 648, 9;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A019576. See A180281 for the case when the balls are indistinguishable.
Rows sums give A000312.
Cf. A245687.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1, k)/j!, j=0..min(k, n))))
    end:
T:= (n, k)-> n!* (b(n$2, k) -b(n$2, k-1)):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Jul 29 2014

f[0, , b] := Boole[b == 0]; f[n_, k_, b_] := f[n, k, b] = Sum[ Binomial[b, i]*f[n - 1, k, b - i], {i, 0, Min[k, b]}]; t[n_, k_] := f[n, k, n] - f[n, k - 1, n]; Flatten[ Table[ t[n, k], {n, 1, 9}, {k, 1, n}]] (* Jean-François Alcover, Mar 09 2012, after Robert Gerbicz *)

/*setup memoization table for args <= M. Could be done dynamically inside f() */
M=10;F=vector(M,i,vector(M,i,vector(M)));
f(n,k,b)={ (!n||!b||!k) & return(!b); F[n][k][b] & return(F[n][k][b]);
F[n][k][b]=sum(i=0,min(k,b),binomial(b,i)*f(n-1,k,b-i)) }
T(n,k)=f(n,k,n)-f(n,k-1,n)
for(n=1,9,print(vector(n,k,T(n,k))))
\\ M. F. Hasler, Aug 19 2010; Based on Robert Gerbicz's code I suggest the following (very naively) memoized version of "f"

A019575(x, z) = Sum ( A049009(p)) where x = A036042(p), z = A049085(p) - Alford Arnold.
From Robert Gerbicz, Aug 19 2010: (Start)
Let f(n,k,b) = number of ways to place b balls to n boxes, where the max in any box is not larger than k. Then T(n,k) = f(n,k,n) - f(n,k-1,n). We have:
f(n, k, b) = if(n=0, if(b=0, 1, 0), Sum_{i=0..min(k, b)} binomial(b, i)*f(n-1, k, b-i)).
T(n,k) = f(n,k,n) - f(n,k-1,n). (End)

Edited by N. J. A. Sloane, Sep 06 2010

A211983 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 19 2012

The order of the partitions of the odd integers is the same as A211999. The order of the partitions of the even integers is the same as A211989.

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

Rows sums give A036042, n>=1.
Other versions are A211984, A211989, A211999. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A211984 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 19 2012

The order of the partitions of the odd integers is the same as A211989. The order of the partitions of the even integers is the same as A211999.

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

Rows sums give A036042, n>=1.
Other versions are A211983, A211989, A211999. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A211985 A list of certain compositions which arise from the ordered partitions of the positive integers in which the shells of each integer are arranged as a spiral.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 19 2012

In order to construct this sequence we use the following rules:
- Consider the partitions of positive integers.
- For each positive integer its shells must be arranged in a spiral.
- The sequence lists one spiral for each positive integer.
- If the integer j is odd then we use the same spiral of A211995.
- If the integer j is even then the first composition listed of each spiral is j.

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

Rows sums give A036042, n>=1.
Mirror of A211986. Other spiral versions are A211987, A211988, A211995-A211998. See also A026792, A211983, A211984, A211989, A211992, A211993, A211994, A211999.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A211989 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 18 2012

The sequence lists the partitions of all positive integers. Each row of the irregular array is a partition of j.
At stage 1, we start with 1.
At stage j > 1, we write the partitions of j using the following rules:
First, we write the partitions of j that do not contain 1 as a part, in reverse-lexicographic order, starting with the partition that contains the part of size j.
Second, we copy from this array the partitions of j-1 in descending order, as a mirror image, starting with the partition that contains the part of size j-2 together with the part of size 1. At the end of each new row, we added a part of size 1.

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

Rows sums give A036042, n>=1.
Other versions are A211983, A211984, A211999. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
Cf. A000041, A026905, A135010, A138121, A138137, A138879, A182703, A206437.

A211994 A list of ordered partitions of the positive integers.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 18 2012

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

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

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

A098546 Table read by rows: row n has a term T(n,k) for each of the partition(n) partitions of n. T(n,k) = binomial(n,m) where m is the number of parts.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 6, 4, 1, 5, 10, 10, 10, 10, 5, 1, 6, 15, 15, 15, 20, 20, 20, 15, 15, 6, 1, 7, 21, 21, 21, 35, 35, 35, 35, 35, 35, 35, 21, 21, 7, 1, 8, 28, 28, 28, 28, 56, 56, 56, 56, 56, 70, 70, 70, 70, 70, 56, 56, 56, 28, 28, 8, 1, 9, 36, 36, 36, 36, 84, 84, 84, 84, 84, 84, 84

Offset: 1

Alford Arnold, Sep 14 2004

A035206 and A036038 were used to generate A049009 (Words over signatures). A098346 and A049019 provide another approach to the same end since A098346 times A049019 also yields A049009. (cf. A000312 and A000670).

Partitions are in Abramowitz and Stegun order. - Franklin T. Adams-Watters, Nov 20 2006

			A036042 begins 1 2 2 3 3 3 4 4 4 4 4 ...
A036043 begins 1 1 2 1 2 3 1 2 2 3 4 ...
so a(n) begins 1 2 1 3 3 1 4 6 6 4 1 ...
Table begins
.
1
2 1
3 3  1
4 6  6  4  1
5 10 10 10 10 5  1
6 15 15 20 15 20 15 20 15 6 1
.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A090657, A000041 (row lengths), A098545 (row sums), A036036, A036042, A036043.

Table[Sequence @@
  Map[Function[p, Binomial[n, Length[p]]], IntegerPartitions[n]], {n,
  1, 10}] (* Olivier Gérard, May 07 2024 *)

a(n) = Combin( A036042(n), A036043(n) )

cp's OEIS Frontend

A036041 Number of prime divisors, counted with multiplicity, of prime signature A025487(n); equals size of associated partition.

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A211999 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

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A333486 Length of the n-th reversed integer partition in graded reverse-lexicographic order. Partition lengths of A228531.

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A019575 Place n distinguishable balls in n boxes (in n^n ways); let T(n,k) = number of ways that the maximum in any box is k, for 1 <= k <= n; sequence gives triangle of numbers T(n,k).

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A211983 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

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A211984 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

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A211985 A list of certain compositions which arise from the ordered partitions of the positive integers in which the shells of each integer are arranged as a spiral.

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A211989 A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.

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A211994 A list of ordered partitions of the positive integers.

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A098546 Table read by rows: row n has a term T(n,k) for each of the partition(n) partitions of n. T(n,k) = binomial(n,m) where m is the number of parts.

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