A334302 -id:A334302 - cp's OEIS frontend

A036036 Triangle read by rows in which row n lists all the parts of all reversed partitions of n, sorted first by length and then lexicographically.

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, 1, 2, 4, 1, 3, 3, 2, 2, 3, 1, 1, 1

Offset: 1

N. J. A. Sloane

First differs from A334442 for reversed partitions of 9. Namely, this sequence has (1,4,4) before (2,2,5), while A334442 has (2,2,5) before (1,4,4). - Gus Wiseman, May 07 2020
This is the "Abramowitz and Stegun" ordering of the partitions, referenced in numerous other sequences. The partitions are in reverse order of the conjugates of the partitions in Mathematica order (A080577). Each partition is the conjugate of the corresponding partition in Maple order (A080576). - Franklin T. Adams-Watters, Oct 18 2006
The "Abramowitz and Stegun" ordering of the partitions is the graded reflected colexicographic ordering of the partitions. - Daniel Forgues, Jan 19 2011
The "Abramowitz and Stegun" ordering of partitions has been traced back to C. F. Hindenburg, 1779, in the Knuth reference, p. 38. See the Hindenburg link, pp. 77-5 with the listing of the partitions for n=10.  This is also mentioned in the P. Luschny link. - Wolfdieter Lang, Apr 04 2011
The "Abramowitz and Stegun" order used here means that the partitions of a given number are listed by increasing number of (nonzero) parts, then by increasing lexicographical order with parts in (weakly) indecreasing order. This differs from n=9 on from A334442 which considers reverse lexicographic order of parts in (weakly) decreasing order. - M. F. Hasler, Jul 12 2015, corrected thanks to Gus Wiseman, May 14 2020
This is the Abramowitz-Stegun ordering of reversed partitions (finite weakly increasing sequences of positive integers). The same ordering of non-reversed partitions is A334301. - Gus Wiseman, May 07 2020

			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;
...

Abramowitz and Stegun, Handbook, p. 831, column labeled "pi".
D. Knuth, The Art of Computer Programming, Vol. 4, fascicle 3, 7.2.1.4, Addison-Wesley, 2005.

Franklin T. Adams-Watters, First 20 rows, 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)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions.
C. F. Hindenburg, Infinitinomii Dignitatum Exponentis Indeterminati, Goettingen 1779.
Peter Luschny, Counting with partitions.
OEIS Wiki, Orderings of partitions (a comparison).
Wikiversity, Lexicographic and colexicographic order

Cf. A036037-A036040.
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 A193073 for the graded lexicographic ordering.
See A228100 for the Fenner-Loizou (binary tree) ordering.
The version ignoring length is A026791.
Same as A036037 with partitions reversed.
The lengths of these partitions are A036043.
The number of distinct parts is A103921.
The corresponding ordering of compositions is A124734.
Showing partitions as Heinz numbers gives A185974.
The version for non-reversed partitions is A334301.
Lexicographically ordered reversed partitions are A026791.
Sorting reversed partitions by Heinz number gives A112798.
The version for revlex instead of lex is A334302.
The version for revlex instead of colex is A334442.
Cf. A000041, A211992, A228531, A296774, A334433, A334435, A334436, A334437, A334438, A334439, A334440, A334441.

Join@@Table[Sort[Reverse/@IntegerPartitions[n]],{n,0,8}] (* Gus Wiseman, May 07 2020 *)
- or -
colen[f_,c_]:=OrderedQ[{Reverse[f],Reverse[c]}];
Reverse/@Join@@Table[Sort[IntegerPartitions[n],colen],{n,0,8}] (* Gus Wiseman, May 07 2020 *)

T036036(n,k)=k&&return(T036036(n)[k]);concat(partitions(n))
\\ If 2nd arg "k" is not given, return the n-th row as a vector. Assumes PARI version >= 2.7.1. See A193073 for "hand made" code.
concat(vector(8,n,T036036(n))) \\ to get the "flattened" sequence
\\ M. F. Hasler, Jul 12 2015

Edited by Daniel Forgues, Jan 21 2011
Edited by M. F. Hasler, Jul 12 2015
Name corrected by Gus Wiseman, May 12 2020

A211992 Triangle read by rows in which row n lists the partitions of n in colexicographic order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 18 2012

The order of the partitions of every integer is reversed with respect to A026792. For example: in A026792 the partitions of 3 are listed as [3], [2, 1], [1, 1, 1], however here the partitions of 3 are listed as [1, 1, 1], [2, 1], [3].
Row n has length A006128(n). Row sums give A066186. Right border gives A000027. The equivalent sequence for compositions (ordered partitions) is A228525. - Omar E. Pol, Aug 24 2013
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. The equivalent sequence for partitions as (weakly) increasing lists and lexicographic order is A026791. - Joerg Arndt, Sep 02 2013

			From _Omar E. Pol_, Aug 24 2013: (Start)
Illustration of initial terms:
-----------------------------------------
n      Diagram          Partition
-----------------------------------------
.       _
1      |_|              1;
.       _ _
2      |_| |            1, 1,
2      |_ _|            2;
.       _ _ _
3      |_| | |          1, 1, 1,
3      |_ _| |          2, 1,
3      |_ _ _|          3;
.       _ _ _ _
4      |_| | | |        1, 1, 1, 1,
4      |_ _| | |        2, 1, 1,
4      |_ _ _| |        3, 1,
4      |_ _|   |        2, 2,
4      |_ _ _ _|        4;
.       _ _ _ _ _
5      |_| | | | |      1, 1, 1, 1, 1,
5      |_ _| | | |      2, 1, 1, 1,
5      |_ _ _| | |      3, 1, 1,
5      |_ _|   | |      2, 2, 1,
5      |_ _ _ _| |      4, 1,
5      |_ _ _|   |      3, 2,
5      |_ _ _ _ _|      5;
.       _ _ _ _ _ _
6      |_| | | | | |    1, 1, 1, 1, 1, 1,
6      |_ _| | | | |    2, 1, 1, 1, 1,
6      |_ _ _| | | |    3, 1, 1, 1,
6      |_ _|   | | |    2, 2, 1, 1,
6      |_ _ _ _| | |    4, 1, 1,
6      |_ _ _|   | |    3, 2, 1,
6      |_ _ _ _ _| |    5, 1,
6      |_ _|   |   |    2, 2, 2,
6      |_ _ _ _|   |    4, 2,
6      |_ _ _|     |    3, 3,
6      |_ _ _ _ _ _|    6;
...
Triangle begins:
[1];
[1,1], [2];
[1,1,1], [2,1], [3];
[1,1,1,1], [2,1,1], [3,1], [2,2], [4];
[1,1,1,1,1], [2,1,1,1], [3,1,1], [2,2,1], [4,1], [3,2], [5];
[1,1,1,1,1,1], [2,1,1,1,1], [3,1,1,1], [2,2,1,1], [4,1,1], [3,2,1], [5,1], [2,2,2], [4,2], [3,3], [6];
(End)
From _Gus Wiseman_, May 10 2020: (Start)
The triangle with partitions shown as Heinz numbers (A334437) 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
(End)

Joerg Arndt, Table of n, a(n) for n = 1..10000
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Cf. A026791, A141285, A194446, A228531.
The graded reversed version is A026792.
The length-sensitive refinement is A036037.
The version for reversed partitions is A080576.
Partition lengths are A193173.
Partition maxima are A194546.
Partition minima are A196931.
The version for compositions is A228525.
The Heinz numbers of these partitions are A334437.
Cf. A036036, A080577, A193073, A228100, A296150, A331581, A334301, A334302, A334436, A334439, A334442.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Join@@Table[Sort[IntegerPartitions[n],colex],{n,0,6}] (* Gus Wiseman, May 10 2020 *)

gen_part(n)=
{  /* Generate partitions of n as weakly increasing lists (order is lex): */
    my(ct = 0);
    my(m, pt);
    my(x, y);
    \\ init:
    my( a = vector( n + (n<=1) ) );
    a[1] = 0;  a[2] = n;  m = 2;
    while ( m!=1,
        y = a[m] - 1;
        m -= 1;
        x = a[m] + 1;
        while ( x<=y,
            a[m] = x;
            y = y - x;
            m += 1;
        );
        a[m] = x + y;
        pt = vector(m, j, a[j]);
    /* for A026791 print partition: */
\\        for (j=1, m, print1(pt[j],", ") );
    /* for A211992 print partition as weakly decreasing list (order is colex): */
        forstep (j=m, 1, -1, print1(pt[j],", ") );
        ct += 1;
    );
    return(ct);
}
for(n=1, 10, gen_part(n) );
\\ Joerg Arndt, Sep 02 2013

A193073 Triangle in which n-th row lists all partitions of n, in graded lexicographical ordering.

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

Offset: 1

M. F. Hasler, Jul 15 2011

The partitions of the integer n are sorted in lexicographical order (cf. link: sums are written with terms in decreasing order, then they are sorted in lexicographical (increasing) order), i.e., as [1,1,...,1], [2,1,...,1], [2,2,...], ..., [n].

			First five rows are:
[[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]]
From _Gus Wiseman_, May 08 2020: (Start)
The sequence of all partitions begins:
  ()           (2,2,1)        (5,1)            (5,2)
  (1)          (3,1,1)        (6)              (6,1)
  (1,1)        (3,2)          (1,1,1,1,1,1,1)  (7)
  (2)          (4,1)          (2,1,1,1,1,1)    (1,1,1,1,1,1,1,1)
  (1,1,1)      (5)            (2,2,1,1,1)      (2,1,1,1,1,1,1)
  (2,1)        (1,1,1,1,1,1)  (2,2,2,1)        (2,2,1,1,1,1)
  (3)          (2,1,1,1,1)    (3,1,1,1,1)      (2,2,2,1,1)
  (1,1,1,1)    (2,2,1,1)      (3,2,1,1)        (2,2,2,2)
  (2,1,1)      (2,2,2)        (3,2,2)          (3,1,1,1,1,1)
  (2,2)        (3,1,1,1)      (3,3,1)          (3,2,1,1,1)
  (3,1)        (3,2,1)        (4,1,1,1)        (3,2,2,1)
  (4)          (3,3)          (4,2,1)          (3,3,1,1)
  (1,1,1,1,1)  (4,1,1)        (4,3)            (3,3,2)
  (2,1,1,1)    (4,2)          (5,1,1)          (4,1,1,1,1)
The triangle with partitions shown as Heinz numbers (A334434) 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
(End)

Alois P. Heinz, Rows n = 1..19, flattened
OEIS Wiki, Orderings of partitions.
Wikiversity, Lexicographic and colexicographic order

See A036036 for the Hindenburg (graded reflected colexicographic) ordering (listed in the Abramowitz and Stegun Handbook).
See A036037 for graded colexicographic ordering.
See A080576 for the Maple (graded reflected lexicographic) ordering.
See A080577 for the Mathematica (graded reverse lexicographic) ordering.
See A228100 for the Fenner-Loizou (binary tree) ordering.
A006128 gives row lengths.
Row n has A000041(n) partitions.
The version for reversed (weakly increasing) partitions is A026791.
Lengths of these partitions appear to be A049085.
Taking colex instead of lex gives A211992.
The generalization to compositions is A228351.
Sorting partitions by Heinz number gives A296150.
The length-sensitive refinement is A334301.
The Heinz numbers of these partitions are A334434.
Cf. A066099, A129129, A185974, A228531, A334302, A334433, A334437, A334439.

row[n_] := Flatten[Reverse[Reverse /@ SplitBy[IntegerPartitions[n], Length] ], 1]; Array[row, 19] // Flatten (* Jean-François Alcover, Dec 05 2016 *)
lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Join@@Table[Sort[IntegerPartitions[n],lexsort],{n,0,8}] (* Gus Wiseman, May 08 2020 *)

A193073_row(n)=concat(vecsort(apply(P->Vec(vecsort(P,,4)),partitions(n)))) \\ The two vecsort() are needed since the PARI function (version >= 2.7.1) yields the partitions in Abramowitz-Stegun order: sorted by increasing length, decreasing largest part, then lex order, with parts in increasing order. - M. F. Hasler, Jun 04 2018 [replaced older code from Jul 12 2015]

def p(n, i):
    if n==0 or i==1: return [[1]*n]
    T = [[i] + x for x in p(n-i, i)] if i<=n else []
    return p(n, i-1) + T
A193073 = lambda n: p(n,n)
for n in (1..5): print(A193073(n)) # Peter Luschny, Aug 07 2015

A334439 Irregular triangle whose 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, 2, 1, 1, 1, 1, 4, 3, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 1, 4, 2, 3, 3, 4, 1, 1, 3, 2, 1, 2, 2, 2, 3, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 6, 1, 5, 2, 4, 3, 5, 1, 1

Offset: 0

Gus Wiseman, May 03 2020

First differs from A036037 for partitions of 9. Namely, this sequence has (5,2,2) before (4,4,1), while A036037 has (4,4,1) before (5,2,2).

This is the Abramowitz-Stegun ordering of integer partitions (A334301) except that the finer order is reverse-lexicographic instead of lexicographic. The version for reversed partitions is A334302.

			The sequence of all partitions begins:
  ()      (32)     (21111)   (22111)    (4211)      (63)
  (1)     (311)    (111111)  (211111)   (3311)      (54)
  (2)     (221)    (7)       (1111111)  (3221)      (711)
  (11)    (2111)   (61)      (8)        (2222)      (621)
  (3)     (11111)  (52)      (71)       (41111)     (531)
  (21)    (6)      (43)      (62)       (32111)     (522)
  (111)   (51)     (511)     (53)       (22211)     (441)
  (4)     (42)     (421)     (44)       (311111)    (432)
  (31)    (33)     (331)     (611)      (221111)    (333)
  (22)    (411)    (322)     (521)      (2111111)   (6111)
  (211)   (321)    (4111)    (431)      (11111111)  (5211)
  (1111)  (222)    (3211)    (422)      (9)         (4311)
  (5)     (3111)   (2221)    (332)      (81)        (4221)
  (41)    (2211)   (31111)   (5111)     (72)        (3321)
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)(11)
             (3)(21)(111)
        (4)(31)(22)(211)(1111)
  (5)(41)(32)(311)(221)(2111)(11111)
Showing partitions as their Heinz numbers (see 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

The version for colex instead of revlex is A036037.
Row lengths are A036043.
Ignoring length gives A080577.
Number of distinct elements in row n appears to be A103921(n).
The version for compositions is A296774.
The Abramowitz-Stegun version (sum/length/lex) is A334301.
The version for reversed partitions is A334302.
Taking Heinz numbers gives A334438.
The version with partitions reversed is A334442.
Lexicographically ordered reversed partitions are A026791.
Lexicographically ordered partitions are A193073.
Sorting partitions by Heinz number gives A296150.
Cf. A000041, A036036, A112798, A124734, A129129, A185974, A228100, A228531, A334433, A334435, A334436.

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

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

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, 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).

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

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, 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).

cp's OEIS Frontend

A036036 Triangle read by rows in which row n lists all the parts of all reversed partitions of n, sorted first by length and then lexicographically.

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A211992 Triangle read by rows in which row n lists the partitions of n in colexicographic order.

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A193073 Triangle in which n-th row lists all partitions of n, in graded lexicographical ordering.

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A334439 Irregular triangle whose rows are all integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

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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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