A228531 -id:A228531 - cp's OEIS frontend

A185974 Partitions in Abramowitz-Stegun order A036036 mapped one-to-one to positive integers.

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, 23, 38, 51, 65, 77, 68, 78, 110, 98, 99, 105, 125, 104, 132, 140, 126, 150, 135, 176, 168, 200, 180, 162, 224, 240, 216, 320, 288, 384, 512, 29, 46, 57, 85, 91, 121, 76, 102, 130, 154, 117, 165, 147, 175, 136, 156, 220, 196, 198, 210, 250, 189, 225, 208, 264, 280, 252, 300, 270, 243, 352, 336, 400, 360, 324, 448, 480, 432, 640, 576, 768, 1024

Offset: 0

Wolfdieter Lang, Feb 10 2011

First differs from A334438 (shifted left once) at a(75) = 98, A334438(76) = 99. - Gus Wiseman, May 20 2020
This mapping of the set of all partitions of N >= 0 to {1, 2, 3, ...} (set of natural numbers) is one to one (bijective). The empty partition for N = 0 maps to 1.
A129129 seems to be analogous, except that the partition ordering A080577 is used. This ordering, however, does not care about the number of parts: e.g., 1^2,4 = 4,1^2 comes before 3^2, so a(23)=28 and a(22)=25 are interchanged.
Also Heinz numbers of all reversed integer partitions (finite weakly increasing sequences of positive integers), sorted first by sum, then by length, and finally lexicographically, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The version for non-reversed partitions is A334433. - Gus Wiseman, May 20 2020

			a(22) = 25 = prime(3)^2 because the 22nd partition in A-St order is the 2-part partition (3,3) of N = 6, because A026905(5) = 18 < 22 <= A026905(6) = 29.
a(23) = 28 = prime(1)^2*prime(4) corresponds to the partition 1+1+4 = 4+1+1 with three parts, also of N = 6.
From _Gus Wiseman_, May 20 2020: (Start)
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
As a triangle of reversed partitions we have:
                             0
                            (1)
                          (2)(11)
                        (3)(12)(111)
                   (4)(13)(22)(112)(1111)
             (5)(14)(23)(113)(122)(1112)(11111)
  (6)(15)(24)(33)(114)(123)(222)(1113)(1122)(11112)(111111)
(End)

M. F. Hasler, Table of n, a(n) for n = 0..9295 (up to partitions of 25), Jan 07 2024
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].
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order.

Row lengths are A000041.
The constructive version is A036036.
Also Heinz numbers of the partitions in A036037.
The generalization to compositions is A124734.
The version for non-reversed partitions is A334433.
The non-reversed length-insensitive version is A334434.
The opposite version (sum/length/revlex) is A334435.
Ignoring length gives A334437.
Sorting reversed partitions by Heinz number gives A112798.
Partitions in lexicographic order are A193073.
Partitions in colexicographic order are A211992.
Graded Heinz numbers are A215366.
Cf. A026791, A036043, A056239, A080577, A228531, A296150, A334301, A334302, A334436, A334438, A334439.

Join@@Table[Times@@Prime/@#&/@Sort[Reverse/@IntegerPartitions[n]],{n,0,8}] (* Gus Wiseman, May 21 2020 *)

A185974_row(n)=[vecprod([prime(i)|i<-p])|p<-partitions(n)] \\ below a helper function:
index_of_partition(n)={for(r=0, oo, my(c = numbpart(r)); n >= c || return([r,n+1]); n -= c)}
/* A185974(n,k), 1 <= k <= A000041(n), gives the k-th partition of n >= 0; if k is omitted, A185974(n) return the term of index n of the flattened sequence a(n >= 0).
  This function is used in other sequences (such as A122172) which need to access the n-th partition as listed in A-S order. */
A185974(n, k=index_of_partition(n))=A185974_row(iferr(k[1], E, k=[k,k]; n))[k[2]] \\ (End)

a(n) = Product_{j=1..N(n)} p(j)^e(j), with p(j):=A000040(j) (j-th prime), and the exponent e(j) >= 0 of the part j in the n-th partition written in Abramowitz-Stegun (A-St) order, indicated in A036036. Note that j^0 is not 1 but has to be omitted in the partition. N(n) is the index (argument) of the smallest A026905-number greater than or equal to n (the index of the A026905-ceiling of n).
From Gus Wiseman, May 21 2020: (Start)
A001221(a(n)) = A103921(n).
A001222(a(n)) = A036043(n).
A056239(a(n)) = A036042(n).
A061395(a(n)) = A049085(n).
(End)

Examples edited by M. F. Hasler, Jan 07 2024

A334301 Irregular triangle read by rows where row k is the k-th integer partition, if partitions are sorted first by sum, then by length, and finally lexicographically.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 29 2020

This is the Abramowitz-Stegun ordering of integer partitions when they are read in the usual (weakly decreasing) order. The case of reversed (weakly increasing) partitions is A036036.

			The sequence of all partitions in Abramowitz-Stegun order begins:
  ()      (41)     (21111)   (31111)    (3221)
  (1)     (221)    (111111)  (211111)   (3311)
  (2)     (311)    (7)       (1111111)  (4211)
  (11)    (2111)   (43)      (8)        (5111)
  (3)     (11111)  (52)      (44)       (22211)
  (21)    (6)      (61)      (53)       (32111)
  (111)   (33)     (322)     (62)       (41111)
  (4)     (42)     (331)     (71)       (221111)
  (22)    (51)     (421)     (332)      (311111)
  (31)    (222)    (511)     (422)      (2111111)
  (211)   (321)    (2221)    (431)      (11111111)
  (1111)  (411)    (3211)    (521)      (9)
  (5)     (2211)   (4111)    (611)      (54)
  (32)    (3111)   (22111)   (2222)     (63)
This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows.
                            0
                           (1)
                        (2) (1,1)
                    (3) (2,1) (1,1,1)
            (4) (2,2) (3,1) (2,1,1) (1,1,1,1)
  (5) (3,2) (4,1) (2,2,1) (3,1,1) (2,1,1,1) (1,1,1,1,1)
Showing partitions as their Heinz numbers (see A334433) gives:
   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

Wikiversity, Lexicographic and colexicographic order

Lexicographically ordered reversed partitions are A026791.
The version for reversed partitions (sum/length/lex) is A036036.
Row lengths are A036043.
Reverse-lexicographically ordered partitions are A080577.
The version for compositions is A124734.
Lexicographically ordered partitions are A193073.
Sorting by Heinz number gives A296150, or A112798 for reversed partitions.
Sorting first by sum, then by Heinz number gives A215366.
Reversed partitions under the dual ordering (sum/length/revlex) are A334302.
Taking Heinz numbers gives A334433.
The reverse-lexicographic version is A334439 (not A036037).
Cf. A000041, A048793, A066099, A162247, A211992, A228100, A228351, A228531.

Join@@Table[Sort[IntegerPartitions[n]],{n,0,8}]

A334433 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally 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, Apr 30 2020

First differs from A334435 at a(75) = 99, A334435(75) = 98.
A permutation of the positive integers.
This is the Abramowitz-Stegun ordering of integer partitions when the parts are read in the usual (weakly decreasing) order. The case of reversed (weakly increasing) partitions is A185974.
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 tetrangle:
                  0
                 (1)
               (2)(11)
             (3)(21)(111)
        (4)(22)(31)(211)(1111)
  (5)(32)(41)(221)(311)(2111)(11111)

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

Row lengths are A000041.
Compositions under the same order are A124734 (triangle).
The version for reversed (weakly increasing) partitions is A185974.
The constructive version is A334301.
Ignoring length gives A334434, or A334437 for reversed partitions.
The dual version (sum/length/revlex) is A334438.
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.
Graded reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Graded lexicographically ordered partitions are A193073.
Graded Heinz numbers are A215366.
Sorting partitions by Heinz number gives A296150.
Partitions in increasing-length reverse-lexicographic order (sum/length/revlex) are A334439 (not A036037).
Cf. A026791, A036043, A048793, A056239, A129129, A211992, A228351, A228531, A334302, A334435, A334436.

Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n]],{n,0,8}]

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

A334302 Irregular triangle read by rows where row k is the k-th reversed integer partition, if reversed partitions are 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, 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, 1, 5, 2, 2, 2, 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, 1, 6, 2, 2, 3

Offset: 0

Gus Wiseman, Apr 30 2020

			The sequence of all reversed partitions begins:
  ()         (1,4)        (1,1,1,1,2)
  (1)        (1,2,2)      (1,1,1,1,1,1)
  (2)        (1,1,3)      (7)
  (1,1)      (1,1,1,2)    (3,4)
  (3)        (1,1,1,1,1)  (2,5)
  (1,2)      (6)          (1,6)
  (1,1,1)    (3,3)        (2,2,3)
  (4)        (2,4)        (1,3,3)
  (2,2)      (1,5)        (1,2,4)
  (1,3)      (2,2,2)      (1,1,5)
  (1,1,2)    (1,2,3)      (1,2,2,2)
  (1,1,1,1)  (1,1,4)      (1,1,2,3)
  (5)        (1,1,2,2)    (1,1,1,4)
  (2,3)      (1,1,1,3)    (1,1,1,2,2)
This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows.
                            0
                           (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)
Showing partitions as their Heinz numbers (see A334435) gives:
   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

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

Row lengths are A036043.
Lexicographically ordered reversed partitions are A026791.
The dual ordering (sum/length/lex) of reversed partitions is A036036.
Reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Lexicographically ordered partitions are A193073.
Graded Heinz numbers are A215366.
Ignoring length gives A228531.
Sorting partitions by Heinz number gives A296150.
The version for compositions is A296774.
The dual ordering (sum/length/lex) of non-reversed partitions is A334301.
Taking Heinz numbers gives A334435.
The version for regular (non-reversed) partitions is A334439 (not A036037).
Cf. A000041, A048793, A066099, A080576, A124734, A162247, A211992, A228100, A228351.

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

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).

A228100 Triangle in which n-th row lists all partitions of n, such that partitions of n into m parts appear in lexicographic order previous to the partitions of n into k parts if k < m. (Fenner-Loizou tree.)

Original entry on oeis.org

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

Offset: 1

Peter Luschny, Aug 10 2013

First differs from A193073 at a(58). - Omar E. Pol, Sep 22 2013

The partition lengths appear to be A331581. - Gus Wiseman, May 12 2020

			The sixth row is:
[1, 1, 1, 1, 1, 1]
[2, 1, 1, 1, 1]
[2, 2, 1, 1]
[3, 1, 1, 1]
[2, 2, 2]
[3, 2, 1]
[4, 1, 1]
[3, 3]
[4, 2]
[5, 1]
[6]
From _Gus Wiseman_, May 10 2020: (Start)
The triangle with partitions shown as Heinz numbers (A333485) begins:
    1
    2
    4   3
    8   6   5
   16  12   9  10   7
   32  24  18  20  15  14  11
   64  48  36  40  27  30  28  25  21  22  13
  128  96  72  80  54  60  56  45  50  42  44  35  33  26  17
(End)

T. I. Fenner, G. Loizou: A binary tree representation and related algorithms for generating integer partitions. The Computer J. 23(4), 332-337 (1980)
D. E. Knuth: The Art of Computer Programming. Generating all combinations and partitions, vol. 4, fasc. 3, 7.2.1.4, exercise 10.
K. Yamanaka, Y. Otachi, Sh. Nakano: Efficient enumeration of ordered trees with k leaves. In: WALCOM: Algorithms and Computation, Lecture Notes in Computer Science Volume 5431, 141-150 (2009)
S. Zaks, D. Richards: Generating trees and other combinatorial objects lexicographically. SIAM J. Comput. 8(1), 73-81 (1979)
A. Zoghbi, I. Stojmenovic': Fast algorithms for generating integer partitions. Int. J. Comput. Math. 70, 319-332 (1998)

Alois P. Heinz, rows n = 1..19, flattened
Peter Luschny, Integer Partition Trees
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

See A036036 for the Hindenburg (graded reflected colexicographic) ordering.
See A036037 for the graded colexicographic ordering.
See A080576 for the Maple (graded reflected lexicographic) ordering.
See A080577 for the Mathematica (graded reverse lexicographic) ordering.
See A182937 the Fenner-Loizou (binary tree in preorder traversal) ordering.
See A193073 for the graded lexicographic ordering.
The version for compositions is A296773.
Taking Heinz numbers gives A333485.
Lexicographically ordered reversed partitions are A026791.
Sorting partitions by Heinz number gives A296150, or A112798 for reversed partitions.
Reversed partitions under the (sum/length/revlex) ordering are A334302.
Cf. A000041, A036043, A048793, A211992, A228351, A228531, A331581, A333484, A334301, A334439.

b:= proc(n, i) b(n, i):= `if`(n=0 or i=1, [[1$n]], [b(n, i-1)[],
      `if`(i>n, [], map(x-> [i, x[]], b(n-i, i)))[]])
    end:
T:= n-> map(h-> h[], sort(b(n$2), proc(x, y) local i;
        if nops(x)<>nops(y) then return nops(x)>nops(y) else
        for i to nops(x) do if x[i]<>y[i] then return x[i]Alois P. Heinz, Aug 13 2013

row[n_] := Flatten[Reverse[Sort[#]]& /@ SplitBy[Sort[IntegerPartitions[n] ], Length], 1] // Reverse; Array[row, 8] // Flatten (* Jean-François Alcover, Dec 05 2016 *)
ralensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]>Length[c],OrderedQ[{f,c}]];
Join@@Table[Sort[IntegerPartitions[n],ralensort],{n,0,8}] (* Gus Wiseman, May 10 2020 *)

from collections import deque
def GeneratePartitions(n, visit):
    p = ([], 0, n)
    queue = deque()
    queue.append(p)
    visit(p)
    while len(queue) > 0 :
        (phead, pheadLen, pnum1s) = queue.popleft()
        if pnum1s != 1 :
            head = phead[:pheadLen] + [2]
            q = (head, pheadLen + 1, pnum1s - 2)
            if 1 <= q[2] : queue.append(q)
            visit(q)
        if pheadLen == 1 or (pheadLen > 1 and \
                      (phead[pheadLen - 1] != phead[pheadLen - 2])) :
            head = phead[:pheadLen]
            head[pheadLen - 1] += 1
            q = (head, pheadLen, pnum1s - 1)
            if 1 <= q[2] : queue.append(q)
            visit(q)
def visit(q): print(q[0] + [1 for i in range(q[2])])
for n in (1..7): GeneratePartitions(n, visit)

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).

A103921 Irregular triangle T(n,m) (n >= 0) read by rows: row n lists numbers of distinct parts of partitions of n in Abramowitz-Stegun order.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 3, 2, 3, 1, 2, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 1, 2, 3, 3, 3, 3, 2, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 2, 3

Offset: 0

Wolfdieter Lang, Mar 24 2005

T(n, m) is the number of distinct parts of the m-th partition of n in Abramowitz-Stegun order; n >= 0, m = 1..p(n) = A000041(n).
The row length sequence of this table is p(n)=A000041(n) (number of partitions).
In order to count distinct parts of a partition consider the partition as a set instead of a multiset. E.g., n=6: read [1,1,1,3] as {1,3} and count the elements, here 2.
Rows are the same as the rows of A115623, but in reverse order.
From Wolfdieter Lang, Mar 17 2011: (Start)
The number of 1s in row number n, n >= 1, is tau(n)=A000005(n), the number of divisors of n.
For the proof read off the divisors d(n,j), j=1..tau(n), from row number n of table A027750, and translate them to the tau(n) partitions d(n,1)^(n/d(n,1)), d(n,2)^(n/d(n,2)),..., d(n,tau(n))^(n/d(n,tau(n))).
See a comment by Giovanni Resta under A000005. (End)
From Gus Wiseman, May 20 2020: (Start)
The name is correct if integer partitions are read in reverse, so that the parts are weakly increasing. The non-reversed version is A334440.
Also the number of distinct parts of the n-th integer partition in lexicographic order (A193073).
Differs from the number of distinct parts in the n-th integer partition in (sum/length/revlex) order (A334439). For example, (6,2,2) has two distinct elements, while (1,4,5) has three.
(End)

			Triangle starts:
  0,
  1,
  1, 1,
  1, 2, 1,
  1, 2, 1, 2, 1,
  1, 2, 2, 2, 2, 2, 1,
  1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1,
  1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 1,
  1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 3, 2, 3, 1, 2, 3, 2, 2, 2, 2, 1,
  1, 2, 2, 2, 2, ...
a(5,4)=2 from the fourth partition of 5 in the mentioned order, i.e., (1^2,3), which has two distinct parts, namely 1 and 3.

Robert Price, Table of n, a(n) for n = 0..9295 (first 25 rows).
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
Wolfdieter Lang, First 10 rows.
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row sums are A000070.
Row lengths are A000041.
The lengths of these partitions are A036043.
The maxima of these partitions are A049085.
The version for non-reversed partitions is A334440.
The version for colex instead of lex is (also) A334440.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun order are A036036.
Reverse-lexicographically ordered partitions are A080577.
Compositions in Abramowitz-Stegun order are A124734.
Cf. A001221, A036037, A112798, A115621, A115623, A185974, A193073, A228531, A334301, A334302, A334433, A334435, A334441.

Join@@Table[Length/@Union/@Sort[Reverse/@IntegerPartitions[n]],{n,0,8}] (* Gus Wiseman, May 20 2020 *)

a(n) = A001221(A185974(n)). - Gus Wiseman, May 20 2020

Edited by Franklin T. Adams-Watters, May 29 2006

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).

cp's OEIS Frontend

A185974 Partitions in Abramowitz-Stegun order A036036 mapped one-to-one to positive integers.

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A334301 Irregular triangle read by rows where row k is the k-th integer partition, if partitions are sorted first by sum, then by length, and finally lexicographically.

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A334433 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally lexicographically.

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A334302 Irregular triangle read by rows where row k is the k-th reversed integer partition, if reversed partitions are sorted first by sum, then by length, and finally reverse-lexicographically.

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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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A228100 Triangle in which n-th row lists all partitions of n, such that partitions of n into m parts appear in lexicographic order previous to the partitions of n into k parts if k < m. (Fenner-Loizou tree.)

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