A194548 -id:A194548 - cp's OEIS frontend

A194602 Integer partitions interpreted as binary numbers.

0, 1, 3, 5, 7, 11, 15, 21, 23, 27, 31, 43, 47, 55, 63, 85, 87, 91, 95, 111, 119, 127, 171, 175, 183, 191, 219, 223, 239, 255, 341, 343, 347, 351, 367, 375, 383, 439, 447, 479, 495, 511, 683, 687, 695, 703, 731, 735, 751, 767, 879, 887, 895, 959, 991, 1023, 1365, 1367, 1371, 1375, 1391

Offset: 0

Tilman Piesk, Aug 30 2011

The 2^(n-1) compositions of n correspond to binary numbers, and the partitions of n can be seen as compositions with addends ordered by size, so they also correspond to binary numbers.
The finite sequence for partitions of n (ordered by size) is the beginning of the sequence for partitions of n+1, which leads to an infinite sequence.
From Tilman Piesk, Jan 30 2016: (Start)
It makes sense to regard the positive values as a triangle with row lengths A002865(n) and row numbers n>=2. In this triangle row n contains all partitions of n with non-one addends only. See link "Triangle with Young diagrams".
This sequence contains all binary palindromes with m runs of n ones separated by single zeros. They are ordered in the array A249544. All the rows and columns of this array are subsequences of this sequence, notably its top row (A000225, the powers of two minus one).
Sequences by Omar E. Pol: The "triangle" A210941 defines the same sequence of partitions. Its n-th row shows the non-one addends of the n-th partition. There are A194548(n) of them, and A141285(n) is the largest among them. (The "triangle" A210941 does not actually form a triangle, but A210941 and A141285 do.) Note that the offset of these sequences is 1 and not 0.
(End)
Numbers whose binary representation has runs of '1's of weakly increasing length (with trailing '0's (introducing a run of length 0) forbidden, i.e., only odd terms beyond 0). - M. F. Hasler, May 14 2020

			From _Joerg Arndt_, Nov 17 2012: (Start)
With leading zeros included, the first A000041(n) terms correspond to the list of partitions of n as nondecreasing compositions in lexicographic order.
For example, the first A000041(10)=42 terms correspond to the partitions of 10 as follows (dots for zeros in the binary expansions):
[ n]   binary(a(n))  a(n)  partition
[ 0]   ..........     0    [ 1 1 1 1 1 1 1 1 1 1 ]
[ 1]   .........1     1    [ 1 1 1 1 1 1 1 1 2 ]
[ 2]   ........11     3    [ 1 1 1 1 1 1 1 3 ]
[ 3]   .......1.1     5    [ 1 1 1 1 1 1 2 2 ]
[ 4]   .......111     7    [ 1 1 1 1 1 1 4 ]
[ 5]   ......1.11    11    [ 1 1 1 1 1 2 3 ]
[ 6]   ......1111    15    [ 1 1 1 1 1 5 ]
[ 7]   .....1.1.1    21    [ 1 1 1 1 2 2 2 ]
[ 8]   .....1.111    23    [ 1 1 1 1 2 4 ]
[ 9]   .....11.11    27    [ 1 1 1 1 3 3 ]
[10]   .....11111    31    [ 1 1 1 1 6 ]
[11]   ....1.1.11    43    [ 1 1 1 2 2 3 ]
[12]   ....1.1111    47    [ 1 1 1 2 5 ]
[13]   ....11.111    55    [ 1 1 1 3 4 ]
[14]   ....111111    63    [ 1 1 1 7 ]
[15]   ...1.1.1.1    85    [ 1 1 2 2 2 2 ]
[16]   ...1.1.111    87    [ 1 1 2 2 4 ]
[17]   ...1.11.11    91    [ 1 1 2 3 3 ]
[18]   ...1.11111    95    [ 1 1 2 6 ]
[19]   ...11.1111   111    [ 1 1 3 5 ]
[20]   ...111.111   119    [ 1 1 4 4 ]
[21]   ...1111111   127    [ 1 1 8 ]
[22]   ..1.1.1.11   171    [ 1 2 2 2 3 ]
[23]   ..1.1.1111   175    [ 1 2 2 5 ]
[24]   ..1.11.111   183    [ 1 2 3 4 ]
[25]   ..1.111111   191    [ 1 2 7 ]
[26]   ..11.11.11   219    [ 1 3 3 3 ]
[27]   ..11.11111   223    [ 1 3 6 ]
[28]   ..111.1111   239    [ 1 4 5 ]
[29]   ..11111111   255    [ 1 9 ]
[30]   .1.1.1.1.1   341    [ 2 2 2 2 2 ]
[31]   .1.1.1.111   343    [ 2 2 2 4 ]
[32]   .1.1.11.11   347    [ 2 2 3 3 ]
[33]   .1.1.11111   351    [ 2 2 6 ]
[34]   .1.11.1111   367    [ 2 3 5 ]
[35]   .1.111.111   375    [ 2 4 4 ]
[36]   .1.1111111   383    [ 2 8 ]
[37]   .11.11.111   439    [ 3 3 4 ]
[38]   .11.111111   447    [ 3 7 ]
[39]   .111.11111   479    [ 4 6 ]
[40]   .1111.1111   495    [ 5 5 ]
[41]   .111111111   511    [ 10 ]
(End)

Tilman Piesk, Table of n, a(n) for n = 0..8348
Tilman Piesk, Same table with binary strings and non-one addends
Tilman Piesk, Triangle with Young diagrams (n = 2..20).
Tilman Piesk, Integer partitions and Permutations and partitions in the OEIS
Tilman Piesk, Python functions, keynum_to_valnum(n) = a(n), valnum_to_keynum(a(n)) = n.
Li-yao Xia, Identities for A194602
Index entries for sequences related to binary expansion of n

Cf. A000041 (partition numbers).
Cf. A002865 (row lengths).
Cf. A002450, A000225 (subsequences).
Cf. A249544 (rows and columns are subsequences).
Cf. A210941, A194548, A141285, A228354, A326956.

lim = 12;
Sort[FromDigits[Reverse@ #, 2] & /@
   Map[If[Length@ # == 0, {0}, Flatten@ Most@ #] &@
     Riffle[#, Table[0, Length@ #]] &,
     Map[Table[1, # - 1] &,
       Sort@ IntegerPartitions@ lim /. 1 -> Nothing, {2}]]]
(* Michael De Vlieger, Feb 14 2016 *)

isA194602(n) = if(!n,1,if(!(n%2),0,my(prl=0,rl=0); while(n, if(0==(n%2),if((prl && rl>prl)||0==(n%4), return(0)); prl=rl; rl=0, rl++); n >>= 1); ((0==prl)||(rl<=prl)))); \\ - Antti Karttunen, Dec 06 2021

a( A000041(n)-1 ) = A000225(n-1) for n>=1. - Tilman Piesk, Apr 16 2012
a( A000041(2n-1) ) = A002450(n)  for n>=1. - Tilman Piesk, Apr 16 2012
a( A249543 ) = A249544. - Tilman Piesk, Oct 31 2014
a(n) = A228354(1+n) - 1. - Antti Karttunen, Dec 06 2021

Comments edited by Li-yao Xia, May 13 2014

Incorrect PARI-program removed by Antti Karttunen, Dec 09 2021

A194547 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n, with partitions in lexicographic order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 10 2011

Row n has length A000041(n). The sum of row n is equal to zero.

			Written as a triangle:
  0;
  -1,1;
  -2,0,2;
  -3,-1,1,0,3;
  -4,-2,0,-1,2,1,4;
  -5,-3,-1,-2,1,0,3,-1,2,1,5;
  -6,-4,-2,-3,0,-1,2,-2,1,0,4,0,3,2,6;
  -7,-5,-3,-4,-1,-2,1,-3,0,-1,3,-1,2,1,5,-2,1,0,4,3,2,7;

Alois P. Heinz, Rows n = 1..26, flattened

Cf. A105805, A135010, A138121, A194546, A194548, A194549.

T:= proc(n) local b, l;
      b:= proc(n, i, t)
            if n=0 then l:=l, i-t
          elif i>n then
          else b(n-i, i, t+1); b(n, i+1, t)
            fi
          end;
      l:= NULL; b(n, 1, 0); l
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Dec 22 2011

T[n_] := Module[{b, l}, b[n0_, i_, t_] := If [n0==0, l = Append[l, i-t], If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]]; l = {}; b[n, 1, 0]; l];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

a(n) = A194546(n) - A193173(n).

More terms from Alois P. Heinz, Dec 22 2011

A194549 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 11 2011

			Written as a triangle:
  1;
  1;
  2;
  0,3;
  1,4;
  -1,2,1,5;
  0,3,2,6;
  -2,1,0,4,3,2,7;
  -1,2,1,5,0,4,3,8;
  -3,0,-1,3,2,1,6,1,5,4,3,9;
  -2,1,0,4,-1,3,2,7,2,1,6,5,4,10;
  -4,-1,-2,2,1,0,5,0,4,3,2,8,-1,3,2,7,1,6,5,4,11;

Alois P. Heinz, Rows n = 1..33, flattened

The sum of row n is A000041(n-1). Row n has length A187219(n).

Cf. A002865, A135010, A138121, A194546, A194547, A194548.

T:= proc(n) local b, l;
      b:= proc(n, i, t)
            if n=0 then l:=l, i-t
          elif i>n then
          else b(n-i, i, t+1); b(n, i+1, t)
            fi
          end;
      if n<2 then 1 else l:= NULL; b(n, 2, 0); l fi
    end:
seq(T(n), n=1..13); # Alois P. Heinz, Dec 20 2011

T[n_] := Module[{b, l}, b[n0_, i_, t_] :=
     If[n0 == 0, l = Append[l, i-t],
     If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]];
     If[n<2, {1}, l = {}; b[n, 2, 0]; l]];
Table[T[n], {n, 1, 13}]  // Flatten (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

a(n) = A141285(n) - A194548(n).

More terms from Alois P. Heinz, Dec 20 2011

A194546 Triangle read by rows: T(n,k) is the largest part of the k-th partition of n, with partitions in colexicographic order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 10 2011

Row n lists the first A000041(n) terms of A141285.

The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is co-lexicographic, see example. - Joerg Arndt, Sep 13 2013

			For n = 5 the partitions of 5 in colexicographic order are:
  1+1+1+1+1
  2+1+1+1
  3+1+1
  2+2+1
  4+1
  3+2
  5
so the fifth row is the largest in each partition: 1,2,3,2,4,3,5
Triangle begins:
  1;
  1,2;
  1,2,3;
  1,2,3,2,4;
  1,2,3,2,4,3,5;
  1,2,3,2,4,3,5,2,4,3,6;
  1,2,3,2,4,3,5,2,4,3,6,3,5,4,7;
  1,2,3,2,4,3,5,2,4,3,6,3,5,4,7,2,4,3,6,5,4,8;
...

Wikiversity, Lexicographic and colexicographic order

The sum of row n is A006128(n).
Cf. A135010, A138121, A141285, A194547, A194548, A194549.
Row lengths are A000041.
Let y be the n-th integer partition in colexicographic order (A211992):
- The maximum of y is a(n).
- The length of y is A193173(n).
- The minimum of y is A196931(n).
- The Heinz number of y is A334437(n).
Lexicographically ordered reversed partitions are A026791.
Reverse-colexicographically ordered partitions are A026792.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
Reverse-lexicographically ordered partitions are A080577.
Lexicographically ordered partitions are A193073.
Cf. A036037, A063008, A115623, A228100, A228531, A238966, A334301.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Max/@Join@@Table[Sort[IntegerPartitions[n],colex],{n,8}] (* Gus Wiseman, May 31 2020 *)

a(n) = A061395(A334437(n)). - Gus Wiseman, May 31 2020

Definition corrected by Omar E. Pol, Sep 12 2013

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

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Feb 20 2012

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

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

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

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

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

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

A207035 Sum of all parts minus the total number of parts of the last section of the set of partitions of n.

Original entry on oeis.org

0, 1, 2, 5, 7, 16, 20, 39, 52, 86, 113, 184, 232, 353, 462, 661, 851, 1202, 1526, 2098, 2670, 3565, 4514, 5967, 7473, 9715, 12162, 15583, 19373, 24625, 30410, 38274, 47112, 58725, 71951, 89129, 108599, 133612, 162259, 198346, 239825, 291718, 351269, 425102

Offset: 1

Omar E. Pol, Feb 20 2012

nonn

			For n = 7 the last section of the set of partitions of 7 looks like this:
.
.        (. . . . . . 7)
.        (. . . 4 . . 3)
.        (. . . . 5 . 2)
.        (. . 3 . 2 . 2)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.
The sum of all parts = 7+4+3+5+2+3+2+2+1*11 = 39, on the other hand the total number of parts is 1+2+2+3+1*11 = 19, so a(7) = 39 - 19 = 20. Note that the number of dots in the picture is also equal to a(7) = 6+5+5+4 = 20.

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

Row sums of triangle A207034. Partial sums give A196087.

Cf. A006128, A066186, A135010, A138121, A138135, A138137, A138879, A138880, A187219, A194548, A207038.

b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0]
    elif i<2 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]*(i-1)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..50);  # Alois P. Heinz, Feb 20 2012

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n==0, {1, 0}, i<2, {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]]*(i-1)}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 13 2015, after Alois P. Heinz *)

a(n) = A138879(n) - A138137(n) = A138880(n) - A138135(n). - Omar E. Pol, Apr 21 2012

G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k)^2 / Product_{j>=2} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021

More terms from Alois P. Heinz, Feb 20 2012

A207038 Partial sums of A207034.

Original entry on oeis.org

0, 1, 3, 5, 8, 11, 15, 18, 22, 26, 31, 35, 40, 45, 51, 55, 60, 65, 71, 77, 83, 90, 95, 101, 107, 114, 120, 127, 134, 142, 147, 153, 159, 166, 173, 180, 188, 195, 203, 211, 219, 228, 234, 241, 248, 256, 263, 271, 279, 288, 296, 304, 313, 322, 331, 341

Offset: 1

Omar E. Pol, Feb 21 2012

			Written as a triangle:
0;
1;
3;
5,8;
11,15;
18,22,26,31;
35,40,45,51;
55,60,65,71,77,83,90;
95,101,107,114,120,127,134,142;
147,153,159,166,173,180,188,195,203,211,219,228;
234,241,248,256,263,271,279,288,296,304,313,322,331,341;

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

Row n has length A187219(n). Right border gives A196087.

Cf. A006128, A135010, A138121, A194548, A207035.

A210941 Triangle read by rows in which row n lists the parts > 1 of the n-th zone of the shell model of partitions, with a(1) = 1.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Apr 18 2012

The n-th zone of the shell model of partitions is formed by the parts of row n followed by infinitely many parts of size 1 (see example and also A210943).

Row n lists the largest part and the parts > 1 of the n-th zone of the model.

			Triangle                First 15 zones of the
begins                  shell model of partitions
--------------------------------------------------
1;                      1 1 1 1 1 1 1 1 1 1 1...
2;                      . 2 1 1 1 1 1 1 1 1 1...
3;                      . . 3 1 1 1 1 1 1 1 1...
2, 2;                   . 2 . 2 1 1 1 1 1 1 1...
4;                      . . . 4 1 1 1 1 1 1 1...
3, 2;                   . . 3 . 2 1 1 1 1 1 1...
5;                      . . . . 5 1 1 1 1 1 1...
2, 2, 2;                . 2 . 2 . 2 1 1 1 1 1...
4, 2;                   . . . 4 . 2 1 1 1 1 1...
3, 3;                   . . 3 . . 3 1 1 1 1 1...
6;                      . . . . . 6 1 1 1 1 1...
3, 2, 2;                . . 3 . 2 . 2 1 1 1 1...
5, 2;                   . . . . 5 . 2 1 1 1 1...
4, 3;                   . . . 4 . . 3 1 1 1 1...
7;                      . . . . . . 7 1 1 1 1...

Omar E. Pol, Illustration of the partitions and regions of n, for n = 1..12
Mikhail Kurkov, PARI program. [verification needed]

Column 1 is A141285. Row n has length A194548(n), n > 1.

Cf. A135010, A138121, A182703, A194711, A206437, A210942, A210943.

a210941(n)={
    my(p=[],r=[1]);
    if(n>1,
    my(c=2);
    while(#r1]));
        c++));
    return(r[1..n])
} \\ Joe Slater, Sep 02 2024

cp's OEIS Frontend

A194602 Integer partitions interpreted as binary numbers.

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A194547 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n, with partitions in lexicographic order.

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A194549 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.

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A194546 Triangle read by rows: T(n,k) is the largest part of the k-th partition of n, with partitions in colexicographic order.

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

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A207035 Sum of all parts minus the total number of parts of the last section of the set of partitions of n.

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A207038 Partial sums of A207034.

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A210941 Triangle read by rows in which row n lists the parts > 1 of the n-th zone of the shell model of partitions, with a(1) = 1.