A129129 -id:A129129 - cp's OEIS frontend

A228531 Triangle read by rows in which row n lists the partitions of n in reverse lexicographic order.

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

Offset: 1

Omar E. Pol, Aug 30 2013

The representation of the partitions (for fixed n) is as (weakly) increasing lists of parts, the order between individual partitions (for the same n) is (list-)reversed lexicographic; see examples. [Joerg Arndt, Sep 03 2013]
Also compositions in the triangle of A066099 that are in nondecreasing order.
The equivalent sequence for compositions (ordered partitions) is A066099.
Row n has length A006128(n).
Row sums give A066186.

			Illustration of initial terms:
---------------------------------
.                    Ordered
n  j     Diagram     partition
---------------------------------
.              _
1  1          |_|    1;
.            _ _
2  1        |  _|    2,
2  2        |_|_|    1, 1;
.          _ _ _
3  1      |  _ _|    3,
3  2      | |  _|    1, 2,
3  3      |_|_|_|    1, 1, 1;
.        _ _ _ _
4  1    |    _ _|    4,
4  2    |  _|_ _|    2, 2,
4  3    | |  _ _|    1, 3,
4  4    | | |  _|    1, 1, 2,
4  5    |_|_|_|_|    1, 1, 1, 1;
.
Triangle begins:
[1];
[2],[1,1];
[3],[1,2],[1,1,1];
[4],[2,2],[1,3],[1,1,2],[1,1,1,1];
[5],[2,3],[1,4],[1,2,2],[1,1,3],[1,1,1,2],[1,1,1,1,1];
[6],[3,3],[2,4],[2,2,2],[1,5],[1,2,3],[1,1,4],[1,1,2,2],[1,1,1,3],[1,1,1,1,2],[1,1,1,1,1,1];
[7],[3,4],[2,5],[2,2,3],[1,6],[1,3,3],[1,2,4],[1,2,2,2],[1,1,5],[1,1,2,3],[1,1,1,4],[1,1,1,2,2],[1,1,1,1,3],[1,1,1,1,1,2],[1,1,1,1,1,1,1];
...

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

Row lengths are A000041.
Partition sums are A036042.
Partition minima are A182715.
Partition lengths are A333486.
The lexicographic version (sum/lex) is A026791.
Compositions under the same order (sum/revlex) are A066099.
The colexicographic version (sum/colex) is A080576.
The version for non-reversed partitions is A080577.
The length-sensitive version (sum/length/revlex) is A334302.
The Heinz numbers of these partitions are A334436.
Partitions in colexicographic order (sum/colex) are A211992.
Partitions in lexicographic order (sum/lex) are A193073.
Cf. A026792, A036036, A049085, A103921, A112798, A115623, A129129, A228351, A331581, A334435, A334439, A334442.

revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
Join@@Table[Sort[Reverse/@IntegerPartitions[n],revlexsort],{n,0,8}] (* Gus Wiseman, May 23 2020 *)

A080576 Triangle in which n-th row lists all partitions of n, in graded reflected lexicographic order.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 23 2003

The graded reflected lexicographic ordering of the partitions is used by Maple. - Daniel Forgues, Jan 19 2011
Each partition here is the conjugate of the corresponding partition in Abramowitz and Stegun order (A036036). The partitions are in the reverse of the order of the partitions in Mathematica order (A080577). - Franklin T. Adams-Watters, Oct 18 2006
Reversing all partitions gives A193073 (the non-reflected version). The version for reversed (weakly increasing) partitions is A211992. Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036. - Gus Wiseman, May 20 2020
Also reversed integer partitions in colexicographic order, cf. A228531. - Gus Wiseman, May 31 2020

			First five rows are:
[[1]]
[[1, 1], [2]]
[[1, 1, 1], [1, 2], [3]]
[[1, 1, 1, 1], [1, 1, 2], [2, 2], [1, 3], [4]]
[[1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 2, 2], [1, 1, 3], [2, 3], [1, 4], [5]]
From _Gus Wiseman_, May 20 2020: (Start)
The sequence of all reversed partitions begins:
  ()       (122)     (15)       (25)
  (1)      (113)     (6)        (16)
  (11)     (23)      (1111111)  (7)
  (2)      (14)      (111112)   (11111111)
  (111)    (5)       (11122)    (1111112)
  (12)     (111111)  (1222)     (111122)
  (3)      (11112)   (11113)    (11222)
  (1111)   (1122)    (1123)     (2222)
  (112)    (222)     (223)      (111113)
  (22)     (1113)    (133)      (11123)
  (13)     (123)     (1114)     (1223)
  (4)      (33)      (124)      (1133)
  (11111)  (114)     (34)       (233)
  (1112)   (24)      (115)      (11114)
(End)

Alois P. Heinz, Rows n = 1..20, flattened
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]. (uses Flash)
A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions
Wikiversity, Lexicographic and colexicographic order

See A080577 for the Mathematica (graded reverse lexicographic) ordering.
See A036036 for the Hindenburg (graded reflected colexicographic) ordering (listed in the Abramowitz and Stegun Handbook).
See A036037 for the graded colexicographic ordering.
See A193073 for the graded lexicographic ordering. - M. F. Hasler, Jul 16 2011
See A228100 for the Fenner-Loizou (binary tree) ordering.
Row n has A000041(n) partitions.
Taking colexicographic instead of lexicographic gives A026791.
Lengths of these partitions appear to be A049085.
Reversing all partitions gives A193073 (the non-reflected version).
The version for reversed (weakly increasing) partitions is A211992.
The generalization to compositions is A228525.
The Heinz numbers of these partitions are A334434.
Cf. A026791, A036037, A112798, A129129, A185974, A228351, A228531, A334301, A334302, A334433, A334437.

Maple
```
with(combinat); partition(6);
```

row[n_] := Flatten[Reverse /@ Reverse[SplitBy[Reverse /@ IntegerPartitions[n], Length]], 1]; Array[row, 7] // Flatten (* Jean-François Alcover, Dec 05 2016 *)
lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Reverse/@Join@@Table[Sort[IntegerPartitions[n],lexsort],{n,0,8}] (* Gus Wiseman, May 20 2020 *)

Edited by Daniel Forgues, Jan 21 2011

A334435 Heinz numbers of all reversed integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 15, 14, 18, 20, 24, 32, 13, 25, 21, 22, 27, 30, 28, 36, 40, 48, 64, 17, 35, 33, 26, 45, 50, 42, 44, 54, 60, 56, 72, 80, 96, 128, 19, 49, 55, 39, 34, 75, 63, 70, 66, 52, 81, 90, 100, 84, 88, 108, 120, 112, 144, 160, 192, 256

Offset: 0

Gus Wiseman, May 02 2020

First differs from A334433 at a(75) = 99, A334433(75) = 98.
First differs from A334436 at a(22) = 22, A334436(22) = 27.
A permutation of the positive integers.
Reversed integer partitions are finite weakly increasing sequences of positive integers.
This is the Abramowitz-Stegun ordering of reversed partitions (A185974) except that the finer order is reverse-lexicographic instead of lexicographic. The version for non-reversed partitions is A334438.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.

			The sequence of terms together with their prime indices begins:
    1: {}            32: {1,1,1,1,1}       42: {1,2,4}
    2: {1}           13: {6}               44: {1,1,5}
    3: {2}           25: {3,3}             54: {1,2,2,2}
    4: {1,1}         21: {2,4}             60: {1,1,2,3}
    5: {3}           22: {1,5}             56: {1,1,1,4}
    6: {1,2}         27: {2,2,2}           72: {1,1,1,2,2}
    8: {1,1,1}       30: {1,2,3}           80: {1,1,1,1,3}
    7: {4}           28: {1,1,4}           96: {1,1,1,1,1,2}
    9: {2,2}         36: {1,1,2,2}        128: {1,1,1,1,1,1,1}
   10: {1,3}         40: {1,1,1,3}         19: {8}
   12: {1,1,2}       48: {1,1,1,1,2}       49: {4,4}
   16: {1,1,1,1}     64: {1,1,1,1,1,1}     55: {3,5}
   11: {5}           17: {7}               39: {2,6}
   15: {2,3}         35: {3,4}             34: {1,7}
   14: {1,4}         33: {2,5}             75: {2,3,3}
   18: {1,2,2}       26: {1,6}             63: {2,2,4}
   20: {1,1,3}       45: {2,2,3}           70: {1,3,4}
   24: {1,1,1,2}     50: {1,3,3}           66: {1,2,5}
Triangle begins:
   1
   2
   3   4
   5   6   8
   7   9  10  12  16
  11  15  14  18  20  24  32
  13  25  21  22  27  30  28  36  40  48  64
  17  35  33  26  45  50  42  44  54  60  56  72  80  96 128
This corresponds to the following tetrangle:
                  0
                 (1)
               (2)(11)
             (3)(12)(111)
        (4)(22)(13)(112)(1111)
  (5)(23)(14)(122)(113)(1112)(11111)

Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The dual version (sum/length/lex) is A185974.
Compositions under the same order are A296774 (triangle).
The constructive version is A334302.
Ignoring length gives A334436.
The version for non-reversed partitions is A334438.
Partitions in this order (sum/length/revlex) are A334439.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colex order (sum/length/colex) are A036037.
Reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Graded lexicographically ordered partitions are A193073.
Partitions in colexicographic (sum/colex) order are A211992.
Graded Heinz numbers are given by A215366.
Sorting partitions by Heinz number gives A296150.
Cf. A056239, A124734, A129129, A228100, A228531, A333219, A333220, A334301, A334433, A334434, A334437.

revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]

A001222(a(n)) = A036043(n).

A334438 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 10, 9, 12, 16, 11, 14, 15, 20, 18, 24, 32, 13, 22, 21, 25, 28, 30, 27, 40, 36, 48, 64, 17, 26, 33, 35, 44, 42, 50, 45, 56, 60, 54, 80, 72, 96, 128, 19, 34, 39, 55, 49, 52, 66, 70, 63, 75, 88, 84, 100, 90, 81, 112, 120, 108, 160, 144, 192, 256

Offset: 0

Gus Wiseman, May 03 2020

First differs from A185974 shifted left once at a(76) = 99, A185974(75) = 98.
A permutation of the positive integers.
This is the Abramowitz-Stegun ordering of integer partitions (A334433) except that the finer order is reverse-lexicographic instead of lexicographic. The version for reversed partitions is A334435.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.

			The sequence of terms together with their prime indices begins:
    1: {}            32: {1,1,1,1,1}       50: {1,3,3}
    2: {1}           13: {6}               45: {2,2,3}
    3: {2}           22: {1,5}             56: {1,1,1,4}
    4: {1,1}         21: {2,4}             60: {1,1,2,3}
    5: {3}           25: {3,3}             54: {1,2,2,2}
    6: {1,2}         28: {1,1,4}           80: {1,1,1,1,3}
    8: {1,1,1}       30: {1,2,3}           72: {1,1,1,2,2}
    7: {4}           27: {2,2,2}           96: {1,1,1,1,1,2}
   10: {1,3}         40: {1,1,1,3}        128: {1,1,1,1,1,1,1}
    9: {2,2}         36: {1,1,2,2}         19: {8}
   12: {1,1,2}       48: {1,1,1,1,2}       34: {1,7}
   16: {1,1,1,1}     64: {1,1,1,1,1,1}     39: {2,6}
   11: {5}           17: {7}               55: {3,5}
   14: {1,4}         26: {1,6}             49: {4,4}
   15: {2,3}         33: {2,5}             52: {1,1,6}
   20: {1,1,3}       35: {3,4}             66: {1,2,5}
   18: {1,2,2}       44: {1,1,5}           70: {1,3,4}
   24: {1,1,1,2}     42: {1,2,4}           63: {2,2,4}
Triangle begins:
   1
   2
   3   4
   5   6   8
   7  10   9  12  16
  11  14  15  20  18  24  32
  13  22  21  25  28  30  27  40  36  48  64
  17  26  33  35  44  42  50  45  56  60  54  80  72  96 128
This corresponds to the following tetrangle:
                  0
                 (1)
               (2)(11)
             (3)(21)(111)
        (4)(31)(22)(211)(1111)
  (5)(41)(32)(311)(221)(2111)(11111)

Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
Ignoring length gives A129129.
Compositions under the same order are A296774 (triangle).
The dual version (sum/length/lex) is A334433.
The version for reversed partitions is A334435.
The constructive version is A334439 (triangle).
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colexicographic order (sum/length/colex) are A036037.
Reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Graded lexicographically ordered partitions are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Graded Heinz numbers are given by A215366.
Sorting partitions by Heinz number gives A296150.
Cf. A056239, A066099, A124734, A185974, A228100, A228531, A333219, A334301, A334302, A334434, A334436, A334437.

revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]

A001221(a(n)) = A103921(n).

A001222(a(n)) = A036043(n).

A334434 Heinz number of the n-th integer partition in graded lexicographic order.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 5, 16, 12, 9, 10, 7, 32, 24, 18, 20, 15, 14, 11, 64, 48, 36, 27, 40, 30, 25, 28, 21, 22, 13, 128, 96, 72, 54, 80, 60, 45, 50, 56, 42, 35, 44, 33, 26, 17, 256, 192, 144, 108, 81, 160, 120, 90, 100, 75, 112, 84, 63, 70, 49, 88, 66, 55, 52, 39, 34, 19

Offset: 0

Gus Wiseman, May 01 2020

A permutation of the positive integers.
This is the graded reverse of the so-called "Mathematica" order (A080577, A129129).
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
As a triangle with row lengths A000041, the sequence starts {{1},{2},{4,3},{8,6,5},...}, so offset is 0.

			The sequence of terms together with their prime indices begins:
    1: {}              11: {5}                 45: {2,2,3}
    2: {1}             64: {1,1,1,1,1,1}       50: {1,3,3}
    4: {1,1}           48: {1,1,1,1,2}         56: {1,1,1,4}
    3: {2}             36: {1,1,2,2}           42: {1,2,4}
    8: {1,1,1}         27: {2,2,2}             35: {3,4}
    6: {1,2}           40: {1,1,1,3}           44: {1,1,5}
    5: {3}             30: {1,2,3}             33: {2,5}
   16: {1,1,1,1}       25: {3,3}               26: {1,6}
   12: {1,1,2}         28: {1,1,4}             17: {7}
    9: {2,2}           21: {2,4}              256: {1,1,1,1,1,1,1,1}
   10: {1,3}           22: {1,5}              192: {1,1,1,1,1,1,2}
    7: {4}             13: {6}                144: {1,1,1,1,2,2}
   32: {1,1,1,1,1}    128: {1,1,1,1,1,1,1}    108: {1,1,2,2,2}
   24: {1,1,1,2}       96: {1,1,1,1,1,2}       81: {2,2,2,2}
   18: {1,2,2}         72: {1,1,1,2,2}        160: {1,1,1,1,1,3}
   20: {1,1,3}         54: {1,2,2,2}          120: {1,1,1,2,3}
   15: {2,3}           80: {1,1,1,1,3}         90: {1,2,2,3}
   14: {1,4}           60: {1,1,2,3}          100: {1,1,3,3}
Triangle begins:
    1
    2
    4   3
    8   6   5
   16  12   9  10   7
   32  24  18  20  15  14  11
   64  48  36  27  40  30  25  28  21  22  13
  128  96  72  54  80  60  45  50  56  42  35  44  33  26  17
  ...
This corresponds to the tetrangle:
                  0
                 (1)
               (11)(2)
             (111)(21)(3)
        (1111)(211)(22)(31)(4)
  (11111)(2111)(221)(311)(32)(41)(5)

Alois P. Heinz, Rows n = 0..28, flattened
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The dual version (sum/revlex) is A129129.
The constructive version is A193073.
Compositions under the same order are A228351.
The length-sensitive version is A334433.
The version for reversed (weakly increasing) partitions is A334437.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
Reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Graded Heinz numbers are A215366.
Sorting partitions by Heinz number gives A296150.
Row sums give A145519.
Cf. A036037, A049085, A056239, A066099, A185974, A211992, A228100, A228531, A333219, A334301, A334302, A334435, A334436, A334438, A334439.

T:= n-> map(p-> mul(ithprime(i), i=p), combinat[partition](n))[]:
seq(T(n), n=0..8);  # Alois P. Heinz, Jan 26 2025

lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n],lexsort],{n,0,8}]
- or -
Join@@Table[Times@@Prime/@#&/@Reverse[IntegerPartitions[n]],{n,0,8}]

A001222(a(n)) appears to be A049085(n).

A334436 Heinz numbers of all reversed integer partitions sorted first by sum and then reverse-lexicographically.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 15, 14, 18, 20, 24, 32, 13, 25, 21, 27, 22, 30, 28, 36, 40, 48, 64, 17, 35, 33, 45, 26, 50, 42, 54, 44, 60, 56, 72, 80, 96, 128, 19, 49, 55, 39, 75, 63, 81, 34, 70, 66, 90, 52, 100, 84, 108, 88, 120, 112, 144, 160, 192, 256

Offset: 0

Gus Wiseman, May 02 2020

First differs from A334435 at a(22) = 27, A334435(22) = 22.
A permutation of the positive integers.
Reversed integer partitions are finite weakly increasing sequences of positive integers. For non-reversed partitions, see A129129 and A228531.
This is the so-called "Mathematica" order (A080577).
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
    1: {}            32: {1,1,1,1,1}       42: {1,2,4}
    2: {1}           13: {6}               54: {1,2,2,2}
    3: {2}           25: {3,3}             44: {1,1,5}
    4: {1,1}         21: {2,4}             60: {1,1,2,3}
    5: {3}           27: {2,2,2}           56: {1,1,1,4}
    6: {1,2}         22: {1,5}             72: {1,1,1,2,2}
    8: {1,1,1}       30: {1,2,3}           80: {1,1,1,1,3}
    7: {4}           28: {1,1,4}           96: {1,1,1,1,1,2}
    9: {2,2}         36: {1,1,2,2}        128: {1,1,1,1,1,1,1}
   10: {1,3}         40: {1,1,1,3}         19: {8}
   12: {1,1,2}       48: {1,1,1,1,2}       49: {4,4}
   16: {1,1,1,1}     64: {1,1,1,1,1,1}     55: {3,5}
   11: {5}           17: {7}               39: {2,6}
   15: {2,3}         35: {3,4}             75: {2,3,3}
   14: {1,4}         33: {2,5}             63: {2,2,4}
   18: {1,2,2}       45: {2,2,3}           81: {2,2,2,2}
   20: {1,1,3}       26: {1,6}             34: {1,7}
   24: {1,1,1,2}     50: {1,3,3}           70: {1,3,4}
Triangle begins:
   1
   2
   3   4
   5   6   8
   7   9  10  12  16
  11  15  14  18  20  24  32
  13  25  21  27  22  30  28  36  40  48  64
  17  35  33  45  26  50  42  54  44  60  56  72  80  96 128
This corresponds to the following tetrangle:
                  0
                 (1)
               (2)(11)
             (3)(12)(111)
        (4)(22)(13)(112)(1111)
  (5)(23)(14)(122)(113)(1112)(11111)

Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
Compositions under the same order are A066099 (triangle).
The version for non-reversed partitions is A129129.
The constructive version is A228531.
The lengths of these partitions are A333486.
The length-sensitive version is A334435.
The dual version (sum/lex) is A334437.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colexicographic order (sum/length/colex) are A036037.
Reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Graded lexicographically ordered partitions are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Graded Heinz numbers are A215366.
Sorting partitions by Heinz number gives A296150.
Partitions in dual Abramowitz-Stegun (sum/length/revlex) order are A334439.
Cf. A056239, A124734, A185974, A228100, A333219, A334301, A334302, A334433, A334434, A334438.

lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Table[Times@@Prime/@#&/@Reverse[Sort[Sort/@IntegerPartitions[n],lexsort]],{n,0,8}]

A001222(a(n)) = A333486(n).

A334437 Heinz number of the n-th reversed integer partition in graded lexicographical order.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 5, 16, 12, 10, 9, 7, 32, 24, 20, 18, 14, 15, 11, 64, 48, 40, 36, 28, 30, 22, 27, 21, 25, 13, 128, 96, 80, 72, 56, 60, 44, 54, 42, 50, 26, 45, 33, 35, 17, 256, 192, 160, 144, 112, 120, 88, 108, 84, 100, 52, 90, 66, 70, 34, 81, 63, 75, 39, 55, 49, 19

Offset: 0

Gus Wiseman, May 03 2020

A permutation of the positive integers.
Reversed integer partitions are finite weakly increasing sequences of positive integers. The non-reversed version is A334434.
This is the graded reverse of the so-called "Mathematica" order (A080577, A129129).
The Heinz number of a reversed integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and reversed partitions.
Also Heinz numbers of partitions in colexicographic order (cf. A211992).
As a triangle with row lengths A000041, the sequence starts {{1},{2},{4,3},{8,6,5},...}, so offset is 0.

			The sequence of terms together with their prime indices begins:
    1: {}              11: {5}                 44: {1,1,5}
    2: {1}             64: {1,1,1,1,1,1}       54: {1,2,2,2}
    4: {1,1}           48: {1,1,1,1,2}         42: {1,2,4}
    3: {2}             40: {1,1,1,3}           50: {1,3,3}
    8: {1,1,1}         36: {1,1,2,2}           26: {1,6}
    6: {1,2}           28: {1,1,4}             45: {2,2,3}
    5: {3}             30: {1,2,3}             33: {2,5}
   16: {1,1,1,1}       22: {1,5}               35: {3,4}
   12: {1,1,2}         27: {2,2,2}             17: {7}
   10: {1,3}           21: {2,4}              256: {1,1,1,1,1,1,1,1}
    9: {2,2}           25: {3,3}              192: {1,1,1,1,1,1,2}
    7: {4}             13: {6}                160: {1,1,1,1,1,3}
   32: {1,1,1,1,1}    128: {1,1,1,1,1,1,1}    144: {1,1,1,1,2,2}
   24: {1,1,1,2}       96: {1,1,1,1,1,2}      112: {1,1,1,1,4}
   20: {1,1,3}         80: {1,1,1,1,3}        120: {1,1,1,2,3}
   18: {1,2,2}         72: {1,1,1,2,2}         88: {1,1,1,5}
   14: {1,4}           56: {1,1,1,4}          108: {1,1,2,2,2}
   15: {2,3}           60: {1,1,2,3}           84: {1,1,2,4}
Triangle begins:
    1
    2
    4   3
    8   6   5
   16  12  10   9   7
   32  24  20  18  14  15  11
   64  48  40  36  28  30  22  27  21  25  13
  128  96  80  72  56  60  44  54  42  50  26  45  33  35  17
This corresponds to the following tetrangle:
                  0
                 (1)
               (11)(2)
             (111)(12)(3)
        (1111)(112)(13)(22)(4)
  (11111)(1112)(113)(122)(14)(23)(5)

Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The constructive version is A026791 (triangle).
The length-sensitive version is A185974.
Compositions under the same order are A228351 (triangle).
The version for non-reversed partitions is A334434.
The dual version (sum/revlex) is A334436.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colexicographic order (sum/length/colex) are A036037.
Graded reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Graded lexicographically ordered partitions are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Graded Heinz numbers are given by A215366.
Sorting partitions by Heinz number gives A296150.
Partitions in dual Abramowitz-Stegun (sum/length/revlex) order are A334439.
Cf. A056239, A066099, A129129, A228531, A333219, A333220, A334301, A334302, A334433, A334435, A334438.

lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Table[Times@@Prime/@#&/@Sort[Sort/@IntegerPartitions[n],lexsort],{n,0,8}]

A001222(a(n)) = A193173(n).

A334442 Irregular triangle whose reversed rows are all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 07 2020

First differs from A036036 for reversed partitions of 9. Namely, this sequence has (2,2,5) before (1,4,4), while A036036 has (1,4,4) before (2,2,5).

			The sequence of all partitions begins:
  ()         (2,3)        (1,1,1,1,2)    (1,1,1,2,2)
  (1)        (1,1,3)      (1,1,1,1,1,1)  (1,1,1,1,1,2)
  (2)        (1,2,2)      (7)            (1,1,1,1,1,1,1)
  (1,1)      (1,1,1,2)    (1,6)          (8)
  (3)        (1,1,1,1,1)  (2,5)          (1,7)
  (1,2)      (6)          (3,4)          (2,6)
  (1,1,1)    (1,5)        (1,1,5)        (3,5)
  (4)        (2,4)        (1,2,4)        (4,4)
  (1,3)      (3,3)        (1,3,3)        (1,1,6)
  (2,2)      (1,1,4)      (2,2,3)        (1,2,5)
  (1,1,2)    (1,2,3)      (1,1,1,4)      (1,3,4)
  (1,1,1,1)  (2,2,2)      (1,1,2,3)      (2,2,4)
  (5)        (1,1,1,3)    (1,2,2,2)      (2,3,3)
  (1,4)      (1,1,2,2)    (1,1,1,1,3)    (1,1,1,5)
This sequence can also be interpreted as the following triangle:
                  0
                 (1)
               (2)(11)
             (3)(12)(111)
        (4)(13)(22)(112)(1111)
  (5)(14)(23)(113)(122)(1112)(11111)
Taking Heinz numbers (A334438) gives:
   1
   2
   3   4
   5   6   8
   7  10   9  12  16
  11  14  15  20  18  24  32
  13  22  21  25  28  30  27  40  36  48  64
  17  26  33  35  44  42  50  45  56  60  54  80  72  96 128

Wikiversity, Lexicographic and colexicographic order

Row lengths are A036043.
The version for reversed partitions is A334301.
The version for colex instead of revlex is A334302.
Taking Heinz numbers gives A334438.
The version with rows reversed is A334439.
Ignoring length gives A335122.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colex order (sum/length/colex) are A036037.
Reverse-lexicographically ordered partitions are A080577.
Lexicographically ordered partitions are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Sorting partitions by Heinz number gives A296150.
Cf. A026791, A112798, A124734, A129129, A185974, A228100, A228531, A296774, A334433, A334435, A334436.

revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]

A334442_row(n)=vecsort(partitions(n),p->concat(#p,-Vecrev(p))) \\ Rows of triangle defined in EXAMPLE (all partitions of n). Wrap into [Vec(p)|p<-...] to avoid "Vecsmall". - M. F. Hasler, May 14 2020

A145519 a(n) = Sum_{k=1..n} A145518(n,k).

Original entry on oeis.org

1, 2, 7, 19, 54, 134, 354, 838, 2057, 4794, 11232, 25412, 58075, 128670, 286152, 625829, 1365653, 2941088, 6331146, 13474533, 28642325, 60404681, 127082128, 265712673, 554608226, 1151374963, 2385950536, 4924685252, 10145267212, 20831428273, 42708248451

Offset: 0

Tilman Neumann, Oct 12 2008

nonn

Row sums of A145518.
Also row sums of A129129, A215366.
a(n) = sum of the Heinz numbers of the partitions of n. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the 3 partitions of 3, namely [3], [1,2], and [1,1,1] we get 5, 2*3=6, and 2*2*2=8, respectively; their sum is a(3) = 19. - Emeric Deutsch, Jun 09 2015

Alois P. Heinz, Table of n, a(n) for n = 0..1000
More terms in A145518 and A145519

Cf. A129129, A145518, A215366, A334434.

b:= proc(n, i) option remember; `if`(n=0 or i<2, 2^n,
      add(b(n-i*j, i-1)*ithprime(i)^j, j=0..iquo(n, i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..35);  # Alois P. Heinz, Feb 19 2013

b[n_, i_] := b[n, i] = If[n == 0 || i < 2, 2^n, Sum[b[n-i*j, i-1]*Prime[i]^j, {j, 0, Quotient[n, i]}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

G.f.: 1/Product_{i>=1}(1-prime(i)*x^i). - Vladeta Jovovic, Nov 09 2008

a(n) ~ c * 2^n, where c = Product_{k>=2} 1/(1 - prime(k)/2^k) = 50.412394245500690832088704444961002125578414895935257436317... . - Vaclav Kotesovec, Sep 10 2014, updated Apr 11 2020

a(0) inserted by Alois P. Heinz, Feb 19 2013

A238966 The number of distinct primes in divisor lattice in canonical order.

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

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

After a(0) = 0, this appears to be the same as A128628. - Gus Wiseman, May 24 2020

Also the number of parts in the n-th integer partition in graded reverse-lexicographic order (A080577). - Gus Wiseman, May 24 2020

			Triangle T(n,k) 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, 4, 3, 4, 5, 6;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], 2014.
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row sums are A006128.
Cf. A036043 in canonical order.
Cf. A001221, A063008.
Row lengths are A000041.
The generalization to compositions is A000120.
The sum of the partition is A036042.
The lexicographic version (sum/lex) is A049085.
Partition lengths of A080577.
The partition has A115623 distinct elements.
The Heinz number of the partition is A129129.
The colexicographic version (sum/colex) is A193173.
The maximum of the partition is A331581.
Partitions in lexicographic order (sum/lex) are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Cf. A026792, A036036, A080576, A103921, A112798, A182715, A333486, A334302, A334435, A334436, A334442.

o:= proc(n) option remember; nops(ifactors(n)[2]) end:
b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
    [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> map(x-> o(mul(ithprime(i)^x[i], i=1..nops(x))), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Mar 26 2020

revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
Table[Length/@Sort[IntegerPartitions[n],revlexsort],{n,0,8}] (* Gus Wiseman, May 24 2020 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {Table[1, {n}]}, Join[ Prepend[#, i]& /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
P[n_] := P[n] = Product[Prime[i]^#[[i]], {i, 1, Length[#]}]& /@ b[n, n];
T[n_, k_] := PrimeNu[P[n][[k + 1]]];
Table[T[n, k], {n, 0, 9}, {k, 0, Length[P[n]] - 1}] // Flatten (* Jean-François Alcover, Jan 03 2022, after Alois P. Heinz in A063008 *)

Row(n)={apply(s->#s, vecsort([Vecrev(p) | p<-partitions(n)], , 4))}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Mar 25 2020

T(n,k) = A001221(A063008(n,k)). - Andrew Howroyd, Mar 25 2020

a(n) = A001222(A129129(n)). - Gus Wiseman, May 24 2020

Offset changed and terms a(50) and beyond from Andrew Howroyd, Mar 25 2020

cp's OEIS Frontend

A228531 Triangle read by rows in which row n lists the partitions of n in reverse lexicographic order.

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A080576 Triangle in which n-th row lists all partitions of n, in graded reflected lexicographic order.

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A334435 Heinz numbers of all reversed integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

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A334438 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

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A334434 Heinz number of the n-th integer partition in graded lexicographic order.

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A334436 Heinz numbers of all reversed integer partitions sorted first by sum and then reverse-lexicographically.

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A334437 Heinz number of the n-th reversed integer partition in graded lexicographical order.

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A334442 Irregular triangle whose reversed rows are all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.