A334442 -id:A334442 - 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

A036037 Triangle read by rows in which row n lists all the parts of all the partitions of n, sorted first by length and then colexicographically.

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

Offset: 1

N. J. A. Sloane

First differs from A334439 for partitions of 9. Namely, this sequence has (4,4,1) before (5,2,2), while A334439 has (5,2,2) before (4,4,1). - Gus Wiseman, May 08 2020
This is also a list of all the possible prime signatures of a number, arranged in graded colexicographic ordering. - N. J. A. Sloane, Feb 09 2014
This is also the Abramowitz-Stegun ordering of reversed partitions (A036036) if the partitions are reversed again after sorting. Partitions sorted first by sum and then colexicographically are A211992. - Gus Wiseman, May 08 2020

			First five rows are:
{{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}}
Up to the fifth row, this is exactly the same as the reverse lexicographic ordering A080577. The first row which differs is the sixth one, which reads ((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)). - _M. F. Hasler_, Jan 23 2020
From _Gus Wiseman_, May 08 2020: (Start)
The sequence of all partitions begins:
  ()         (3,2)        (2,1,1,1,1)
  (1)        (3,1,1)      (1,1,1,1,1,1)
  (2)        (2,2,1)      (7)
  (1,1)      (2,1,1,1)    (6,1)
  (3)        (1,1,1,1,1)  (5,2)
  (2,1)      (6)          (4,3)
  (1,1,1)    (5,1)        (5,1,1)
  (4)        (4,2)        (4,2,1)
  (3,1)      (3,3)        (3,3,1)
  (2,2)      (4,1,1)      (3,2,2)
  (2,1,1)    (3,2,1)      (4,1,1,1)
  (1,1,1,1)  (2,2,2)      (3,2,1,1)
  (5)        (3,1,1,1)    (2,2,2,1)
  (4,1)      (2,2,1,1)    (3,1,1,1,1)
(End)

Robert Price, Table of n, a(n) for n = 1..3615, 15 rows.
Wikiversity, Lexicographic and colexicographic order

See A036036 for the graded reflected colexicographic ("Abramowitz and Stegun" or Hindenburg) ordering.
See A080576 for the graded reflected lexicographic ("Maple") ordering.
See A080577 for the graded reverse lexicographic ("Mathematica") ordering: differs from a(48) on!
See A228100 for the Fenner-Loizou (binary tree) ordering.
See also A036038, A036039, A036040: (multinomial coefficients).
Partition lengths are A036043.
Reversing all partitions gives A036036.
The number of distinct parts is A103921.
Taking Heinz numbers gives A185974.
The version ignoring length is A211992.
The version for revlex instead of colex is A334439.
Lexicographically ordered reversed partitions are A026791.
Reverse-lexicographically ordered partitions are A080577.
Sorting partitions by Heinz number gives A296150.
Cf. A000041, A124734, A193073, A228100, A228531, A296774, A334301, A334433, A334436, A334437, A334442.

Reverse/@Join@@Table[Sort[Reverse/@IntegerPartitions[n]],{n,8}] (* Gus Wiseman, May 08 2020 *)
- or -
colen[f_,c_]:=OrderedQ[{Reverse[f],Reverse[c]}];
Join@@Table[Sort[IntegerPartitions[n],colen],{n,8}] (* Gus Wiseman, May 08 2020 *)

Name corrected by Gus Wiseman, May 12 2020

Mathematica programs corrected to reflect offset of one and not zero by Robert Price, Jun 04 2020

A026791 Triangle in which n-th row lists juxtaposed lexicographically ordered partitions of n; e.g., the partitions of 3 (1+1+1,1+2,3) appear as 1,1,1,1,2,3 in row 3.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

Differs from A080576 in a(18): Here, (...,1+3,2+2,4), there (...,2+2,1+3,4).
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 lexicographic (see example). - Joerg Arndt, Sep 03 2013
The equivalent sequence for compositions (ordered partitions) is A228369. - Omar E. Pol, Oct 19 2019

			First six rows are:
[[1]];
[[1, 1], [2]];
[[1, 1, 1], [1, 2], [3]];
[[1, 1, 1, 1], [1, 1, 2], [1, 3], [2, 2], [4]];
[[1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 1, 3], [1, 2, 2], [1, 4], [2, 3], [5]];
[[1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 2], [1, 1, 1, 3], [1, 1, 2, 2], [1, 1, 4], [1, 2, 3], [1, 5], [2, 2, 2], [2, 4], [3, 3], [6]];
...
From _Omar E. Pol_, Sep 03 2013: (Start)
Illustration of initial terms:
----------------------------------
.                     Ordered
n  j      Diagram     partition j
----------------------------------
.               _
1  1           |_|    1;
.             _ _
2  1         | |_|    1, 1,
2  2         |_ _|    2;
.           _ _ _
3  1       | | |_|    1, 1, 1,
3  2       | |_ _|    1, 2,
3  3       |_ _ _|    3;
.         _ _ _ _
4  1     | | | |_|    1, 1, 1, 1,
4  2     | | |_ _|    1, 1, 2,
4  3     | |_ _ _|    1, 3,
4  4     |   |_ _|    2, 2,
4  5     |_ _ _ _|    4;
...
(End)

Alois P. Heinz, Rows n = 1..19, flattened
Wikiversity, Lexicographic and colexicographic order

Row lengths are given in A006128.
Partition lengths are in A193173.
Other partition orderings: A026792, A036037, A080577, A125106, A139100, A181087, A181317, A182937, A228100, A240837, A242628.
Row lengths are A000041.
Partition sums are A036042.
Partition minima are A196931.
Partition maxima are A194546.
The reflected version is A211992.
The length-sensitive version (sum/length/lex) is A036036.
The colexicographic version (sum/colex) is A080576.
The version for non-reversed partitions is A193073.
Compositions under the same ordering (sum/lex) are A228369.
The reverse-lexicographic version (sum/revlex) is A228531.
The Heinz numbers of these partitions are A334437.
Cf. A049085, A103921, A112798, A115623, A129129, A331581, A334435, A334439, A334442.

T:= proc(n) local b, ll;
      b:= proc(n,l)
            if n=0 then ll:= ll, 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..8);  # Alois P. Heinz, Jul 16 2011

T[n0_] := Module[{b, ll}, b[n_, l_] := If[n == 0, ll = Join[ll, l], Table[ b[n - i, Append[l, i]], {i, If[l == {}, 1, l[[-1]]], n}]]; ll = {}; b[n0, {}]; ll]; Table[T[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Aug 05 2015, after Alois P. Heinz *)
Table[DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions[n]], x_ /; x == 0, 2], {n, 7}] // Flatten (* Robert Price, May 18 2020 *)

t = [[[]]]
for n in range(1, 10):
    p = []
    for minp in range(1, n):
        p += [[minp] + pp for pp in t[n-minp] if min(pp) >= minp]
    t.append(p + [[n]])
print(t)
# Andrey Zabolotskiy, Oct 18 2019

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]

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 *)

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

Original entry on oeis.org

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

Offset: 0

Sung-Hyuk Cha, Mar 07 2014

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

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

			Triangle T(n,k) begins:
  0;
  1;
  1, 2;
  1, 2, 3;
  1, 2, 2, 3, 4;
  1, 2, 2, 3, 3, 4, 5;
  1, 2, 2, 3, 2, 3, 4, 3, 4, 5, 6;
  ...

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

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

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

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

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

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

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

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

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}]

A333484 Sort all positive integers, first by sum of prime indices (A056239), then by decreasing number of prime indices (A001222).

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 10 2020

A refinement of A215366.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			Triangle begins:
    1
    2
    4   3
    8   6   5
   16  12   9  10   7
   32  24  18  20  14  15  11
   64  48  36  40  27  28  30  21  22  25  13
  128  96  72  80  54  56  60  42  44  45  50  26  33  35  17

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

Row lengths are A000041.
Ignoring length gives A215366 (graded Heinz numbers).
Sorting by increasing length gives A333483.
Number of prime indices is A001222.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in (sum/length/colex) order are A036037.
Sum of prime indices is A056239.
Reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Lexicographically ordered partitions are A193073.
Sorting partitions by Heinz number gives A296150.
Cf. A124734, A129129, A211992, A228100, A333219, A334301, A334433, A334434, A334439, A334441, A334442.

Join@@@Table[Sort[Times@@Prime/@#&/@IntegerPartitions[n,{k}]],{n,0,8},{k,n,0,-1}]

A333486 Length of the n-th reversed integer partition in graded reverse-lexicographic order. Partition lengths of A228531.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 23 2020

			Triangle begins:
  0
  1
  1 2
  1 2 3
  1 2 2 3 4
  1 2 2 3 3 4 5
  1 2 2 3 2 3 3 4 4 5 6
  1 2 2 3 2 3 3 4 3 4 4 5 5 6 7
  1 2 2 2 3 3 4 2 3 3 4 3 4 4 5 4 5 5 6 6 7 8

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

Row lengths are A000041.
The generalization to compositions is A000120.
Row sums are A006128.
The same partition has sum A036042.
The length-sensitive version (sum/length/revlex) is A036043.
The colexicographic version (sum/colex) is A049085.
The same partition has minimum A182715.
The lexicographic version (sum/lex) is A193173.
The tetrangle of these partitions is A228531.
The version for non-reversed partitions is A238966.
The same partition has Heinz number A334436.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
Partitions in lexicographic order (sum/lex) are A193073.
Partitions in colexicographic order (sum/colex) are A211992.
Partitions in opposite Abramowitz-Stegun order (sum/length/revlex) are A334439.
Cf. A026792, A049085, A080576, A080577, A103921, A112798, A115623, A129129, A331581, A334302, A334435, A334442.

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

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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A036037 Triangle read by rows in which row n lists all the parts of all the partitions of n, sorted first by length and then colexicographically.

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A026791 Triangle in which n-th row lists juxtaposed lexicographically ordered partitions of n; e.g., the partitions of 3 (1+1+1,1+2,3) appear as 1,1,1,1,2,3 in row 3.

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

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

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