A228531 -id:A228531 - cp's OEIS frontend

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

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

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 *)

A331581 Maximum part of the n-th integer partition in graded reverse-lexicographic order (A080577); a(1) = 0.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 08 2020

The first partition ranked by A080577 is (); there is no zeroth partition.

			The sequence of all partitions in graded reverse-lexicographic order begins as follows. The terms are the initial parts.
  ()         (3,2)        (2,1,1,1,1)    (2,2,1,1,1)
  (1)        (3,1,1)      (1,1,1,1,1,1)  (2,1,1,1,1,1)
  (2)        (2,2,1)      (7)            (1,1,1,1,1,1,1)
  (1,1)      (2,1,1,1)    (6,1)          (8)
  (3)        (1,1,1,1,1)  (5,2)          (7,1)
  (2,1)      (6)          (5,1,1)        (6,2)
  (1,1,1)    (5,1)        (4,3)          (6,1,1)
  (4)        (4,2)        (4,2,1)        (5,3)
  (3,1)      (4,1,1)      (4,1,1,1)      (5,2,1)
  (2,2)      (3,3)        (3,3,1)        (5,1,1,1)
  (2,1,1)    (3,2,1)      (3,2,2)        (4,4)
  (1,1,1,1)  (3,1,1,1)    (3,2,1,1)      (4,3,1)
  (5)        (2,2,2)      (3,1,1,1,1)    (4,2,2)
  (4,1)      (2,2,1,1)    (2,2,2,1)      (4,2,1,1)
Triangle begins:
  0
  1
  2 1
  3 2 1
  4 3 2 2 1
  5 4 3 3 2 2 1
  6 5 4 4 3 3 3 2 2 2 1
  7 6 5 5 4 4 4 3 3 3 3 2 2 2 1
  8 7 6 6 5 5 5 4 4 4 4 4 3 3 3 3 3 2 2 2 2 1

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

Row lengths are A000041.
Lexicographically ordered reversed partitions are A026791.
Reverse-colexicographically ordered partitions are A026792.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
The version for compositions is A065120 or A333766.
Reverse-lexicographically ordered partitions are A080577.
Distinct parts of these partitions are counted by A115623.
Lexicographically ordered partitions are A193073.
Colexicographically ordered partitions are A211992.
Lengths of these partitions are A238966.
Cf. A036037, A048793, A063008, A066099, A129129, A185974, A228100, A228531, A334301, A334434, A334436, A334438.

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

a(n) = A061395(A129129(n - 1)).

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}]

A335122 Irregular triangle whose reversed rows are all integer partitions in graded reverse-lexicographic order.

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

Offset: 0

Gus Wiseman, May 24 2020

First differs from A036036 for partitions of 6.
First differs from A334442 for partitions of 6.
Also reversed partitions in reverse-colexicographic order.

			The sequence of all reversed partitions begins:
  ()         (1,1,3)        (7)              (8)
  (1)        (1,2,2)        (1,6)            (1,7)
  (2)        (1,1,1,2)      (2,5)            (2,6)
  (1,1)      (1,1,1,1,1)    (1,1,5)          (1,1,6)
  (3)        (6)            (3,4)            (3,5)
  (1,2)      (1,5)          (1,2,4)          (1,2,5)
  (1,1,1)    (2,4)          (1,1,1,4)        (1,1,1,5)
  (4)        (1,1,4)        (1,3,3)          (4,4)
  (1,3)      (3,3)          (2,2,3)          (1,3,4)
  (2,2)      (1,2,3)        (1,1,2,3)        (2,2,4)
  (1,1,2)    (1,1,1,3)      (1,1,1,1,3)      (1,1,2,4)
  (1,1,1,1)  (2,2,2)        (1,2,2,2)        (1,1,1,1,4)
  (5)        (1,1,2,2)      (1,1,1,2,2)      (2,3,3)
  (1,4)      (1,1,1,1,2)    (1,1,1,1,1,2)    (1,1,3,3)
  (2,3)      (1,1,1,1,1,1)  (1,1,1,1,1,1,1)  (1,2,2,3)
We have the following tetrangle of reversed partitions:
                             0
                            (1)
                          (2)(11)
                        (3)(12)(111)
                   (4)(13)(22)(112)(1111)
             (5)(14)(23)(113)(122)(1112)(11111)
  (6)(15)(24)(114)(33)(123)(1113)(222)(1122)(11112)(111111)

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

Row lengths are A000041.
The version for reversed partitions is A026792.
The version for colex instead of revlex is A026791.
The version for lex instead of revlex is A080576.
The non-reflected version is A080577.
The number of distinct parts is A115623.
Taking Heinz numbers gives A129129.
The version for compositions is A228351.
Partition lengths are A238966.
Partition maxima are A331581.
The length-sensitive version is A334442.
Lexicographically ordered partitions are A193073.
Partitions in colexicographic order are A211992.
Cf. A036036, A036037, A112798, A129129, A228531, A296774, A334301, A334302, A334435, A334436, A334438, A334439.

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

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]]&]

cp's OEIS Frontend

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.

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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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A331581 Maximum part of the n-th integer partition in graded reverse-lexicographic order (A080577); a(1) = 0.

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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.

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