A334437 -id:A334437 - cp's OEIS frontend

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

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).

A193173 Triangle in which n-th row lists the number of elements in lexicographically ordered partitions of n, A026791.

Original entry on oeis.org

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

Offset: 1

Alois P. Heinz, Jul 17 2011

This sequence first differs from A049085 in the partitions of 6 (at flattened index 22):
6, 5, 4, 4, 3, 3, 2, 3, 2, 2, 1 (this sequence);
6, 5, 4, 3, 4, 3, 2, 3, 2, 2, 1 (A049085).
- Jason Kimberley, Oct 27 2011
Rows sums give A006128, n >= 1. - Omar E. Pol, Dec 06 2011
The name is correct if the partitions are read in reverse, so that the parts are weakly increasing. The version for non-reversed partitions is A049085.

			The lexicographically ordered partitions of 3 are [[1, 1, 1], [1, 2], [3]], thus row 3 has 3, 2, 1.
Triangle begins:
  1;
  2, 1;
  3, 2, 1;
  4, 3, 2, 2, 1;
  5, 4, 3, 3, 2, 2, 1;
  6, 5, 4, 4, 3, 3, 2, 3, 2, 2, 1;
  ...

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

Row lengths are A000041.
Partition lengths of A026791.
The version ignoring length is A036043.
The version for non-reversed partitions is A049085.
The maxima of these partitions are A194546.
Reversed partitions in Abramowitz-Stegun order are A036036.
Reverse-lexicographically ordered partitions are A080577.
Cf. A001222, A115623, A129129, A185974, A193073, A211992, A228531, A334302, A334434, A334437, A334440, A334441.

T:= proc(n) local b, ll;
      b:= proc(n,l)
            if n=0 then ll:= ll, nops(l)
            else seq(b(n-i, [l[], i]), i=`if`(l=[], 1, l[-1])..n) fi
          end;
      ll:= NULL; b(n, []); ll
    end:
seq(T(n), n=1..11);

lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Table[Length/@Sort[Reverse/@IntegerPartitions[n],lexsort],{n,0,10}] (* Gus Wiseman, May 22 2020 *)

A344086 Flattened tetrangle of strict integer partitions sorted first by sum, then lexicographically.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 11 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (21)(3)
  4: (31)(4)
  5: (32)(41)(5)
  6: (321)(42)(51)(6)
  7: (421)(43)(52)(61)(7)
  8: (431)(521)(53)(62)(71)(8)
  9: (432)(531)(54)(621)(63)(72)(81)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
Taking revlex instead of lex gives A118457.
The not necessarily strict version is A193073.
The version for reversed partitions is A246688.
The Heinz numbers of these partitions grouped by sum are A246867.
The ordered generalization is A339351.
Taking colex instead of lex gives A344087.
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts reversed strict partitions by Heinz number.
A329631 sorts strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A211992, A228100, A228351, A228531, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A344085, A344086, A344088, A344089.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

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

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

A344089 Flattened tetrangle of reversed strict integer partitions, sorted first by length and then colexicographically.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 12 2021

First differs from the revlex (instead of colex) version for partitions of 12.
The zeroth row contains only the empty partition.
A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (3)(12)
  4: (4)(13)
  5: (5)(23)(14)
  6: (6)(24)(15)(123)
  7: (7)(34)(25)(16)(124)
  8: (8)(35)(26)(17)(134)(125)
  9: (9)(45)(36)(27)(18)(234)(135)(126)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724 plus one.
Taking lex instead of colex gives A026793 (non-reversed: A118457).
Triangle sums are A066189.
Reversing all partitions gives A344090.
The non-strict version is A344091.
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
Cf. A005117, A014466, A209862, A325683, A325859.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A246688, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344087, A344088, A344089.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Table[Reverse/@Sort[Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,30}]

A344085 Triangle of squarefree numbers first grouped by greatest prime factor, then sorted by omega, then in increasing order, read by rows.

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 77, 66, 110, 154, 165, 231, 385, 330, 462, 770, 1155, 2310, 13, 26, 39, 65, 91, 143, 78, 130, 182, 195, 273, 286, 429, 455, 715, 1001, 390, 546, 858, 910, 1365, 1430, 2002, 2145, 3003, 5005, 2730, 4290, 6006, 10010, 15015, 30030

Offset: 1

Gus Wiseman, May 11 2021

Differs from A339195 in having 77 before 66.

			Triangle begins:
   1
   2
   3   6
   5  10  15  30
   7  14  21  35  42  70 105 210

Row lengths are A000079.
Grouping by greatest prime factor only gives A339195.
Row sums are 1 and A339360.
Cf. A001221, A005117, A005183, A014466, A019565, A187769, A209862.
Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A299755, A296774, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344086, A344087, A344088, A344089.
Partition/composition applications: A001793, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124.

nn=4;
GatherBy[SortBy[Select[Range[Times@@Prime/@Range[nn]],SquareFreeQ[#]&&PrimePi[FactorInteger[#][[-1,1]]]<=nn&],PrimeOmega],FactorInteger[#][[-1,1]]&]

A344087 Flattened tetrangle of strict integer partitions sorted first by sum, then colexicographically.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 11 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (21)(3)
  4: (31)(4)
  5: (41)(32)(5)
  6: (321)(51)(42)(6)
  7: (421)(61)(52)(43)(7)
  8: (521)(431)(71)(62)(53)(8)
  9: (621)(531)(81)(432)(72)(63)(54)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
Taking revlex instead of colex gives A118457.
The not necessarily strict version is A211992.
Taking lex instead of colex gives A344086.
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080576, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A246688, A272020, A299755, A296774, A304038, A319247, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344088, A344089, A344091.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Table[Sort[Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]

cp's OEIS Frontend

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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A193173 Triangle in which n-th row lists the number of elements in lexicographically ordered partitions of n, A026791.

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A344086 Flattened tetrangle of strict integer partitions sorted first by sum, then lexicographically.

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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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A344089 Flattened tetrangle of reversed strict integer partitions, sorted first by length and then colexicographically.