A026791 -id:A026791 - cp's OEIS frontend

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

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 *)

A318271 The optimum crossing time for the Bridge and Torch problem, given that the crossing times for the group's members are given by the n-th partition in A026791.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 5, 4, 3, 2, 4, 7, 6, 5, 5, 4, 3, 5, 9, 8, 7, 6, 6, 6, 5, 6, 4, 3, 6, 11, 10, 9, 8, 8, 7, 7, 8, 7, 7, 6, 7, 5, 4, 7, 13, 12, 11, 10, 10, 9, 9, 9, 8, 7, 8, 9, 8, 8, 7, 10, 8, 8, 6, 5, 4, 8, 15, 14, 13, 12, 12, 11, 11, 11, 10, 9, 10, 10, 9, 8, 9

Offset: 1

Peter Kagey, Aug 22 2018

			When the crossing times are [1,2,5,10], the minimum total time for the group to cross is 17 minutes:
  (2m)  1 and 2 cross,
  (1m)  1 returns,
  (10m) 5 and 10 cross,
  (2m)  2 returns,
  (2m)  1 and 2 cross.
+----+--------------------+------+
|  n | Crossing times     | a(n) |
+----+--------------------+------+
|  1 | [1]                |  1   |
|  2 | [1, 1]             |  1   |
|  3 | [2]                |  2   |
|  4 | [1, 1, 1]          |  3   |
|  5 | [1, 2]             |  2   |
|  6 | [3]                |  3   |
|  7 | [1, 1, 1, 1]       |  5   |
|  8 | [1, 1, 2]          |  4   |
|  9 | [1, 3]             |  3   |
| 10 | [2, 2]             |  2   |
| 11 | [4]                |  4   |
| 12 | [1, 1, 1, 1, 1]    |  7   |
| 13 | [1, 1, 1, 2]       |  6   |
| 14 | [1, 1, 3]          |  5   |
| 15 | [1, 2, 2]          |  5   |
| 16 | [1, 4]             |  4   |
| 17 | [2, 3]             |  3   |
| 18 | [5]                |  5   |
| 19 | [1, 1, 1, 1, 1, 1] |  9   |
| 20 | [1, 1, 1, 1, 2]    |  8   |
| 21 | [1, 1, 1, 3]       |  7   |
| 22 | [1, 1, 2, 2]       |  6   |
| 23 | [1, 1, 4]          |  6   |
| 24 | [1, 2, 3]          |  6   |
| 25 | [1, 5]             |  5   |
| 26 | [2, 2, 2]          |  6   |
| 27 | [2, 4]             |  4   |
| 28 | [3, 3]             |  3   |
| 29 | [6]                |  6   |
+----+--------------------+------+

User baseman101, The Bridge and Torch Problem, Programming Puzzles & Code Golf Stack Exchange.
Wikipedia, Bridge and torch problem.

Cf. A026791, A078476.

function BT(p)
    n = length(p)
    p[end] = -(sum(p) + (n > 2 ? (n-3) * p[1] : 0))
    if n >= 3
        q = 2p[2] - p[1]; tog = false
        for k in n-1:-1:1
            (tog = ~tog) && p[k] > q ? p[k] -= q : p[k] = 0
        end
    end
-sum(p) end
[BT(p) for n in 1:9 for p in A026791(n)] |> println # Peter Luschny, Oct 18 2019

Terms a(45) and beyond added using Erwan's program from CodeGolf StackExchange by Andrey Zabolotskiy, Oct 18 2019

A066099 Triangle read by rows, in which row n lists the compositions of n in reverse lexicographic order.

Original entry on oeis.org

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

Offset: 1

Alford Arnold, Dec 30 2001

The representation of the compositions (for fixed n) is as lists of parts, the order between individual compositions (for the same n) is (list-)reversed lexicographic; see the example by Omar E. Pol. - Joerg Arndt, Sep 03 2013
This is the standard ordering for compositions in this database; it is similar to the Mathematica ordering for partitions (A080577). Other composition orderings include A124734 (similar to the Abramowitz & Stegun ordering for partitions, A036036), A108244 (similar to the Maple partition ordering, A080576), etc (see crossrefs).
Factorize each term in A057335; sequence records the values of the resulting exponents. It also runs through all possible permutations of multiset digits.
This can be regarded as a table in two ways: with each composition as a row, or with the compositions of each integer as a row. The first way has A000120 as row lengths and A070939 as row sums; the second has A001792 as row lengths and A001788 as row sums. - Franklin T. Adams-Watters, Nov 06 2006
This sequence includes every finite sequence of positive integers. - Franklin T. Adams-Watters, Nov 06 2006
Compositions (or ordered partitions) are also generated in sequence A101211. - Alford Arnold, Dec 12 2006
The equivalent sequence for partitions is A228531. - Omar E. Pol, Sep 03 2013
The sole partition of zero has no components, not a single component of length one. Hence the first nonempty row is row 1. - Franklin T. Adams-Watters, Apr 02 2014 [Edited by Andrey Zabolotskiy, May 19 2018]
See sequence A261300 for another version where the terms of each composition are concatenated to form one single integer: (0, 1, 2, 11, 3, 21, 12, 111,...). This also shows how the terms can be obtained from the binary numbers A007088, cf. Arnold's first Example. - M. F. Hasler, Aug 29 2015
The k-th composition in the list is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This is described as the standard ordering used in the OEIS, although the sister sequence A228351 is also sometimes considered to be canonical. Both sequences define a bijective correspondence between nonnegative integers and integer compositions. - Gus Wiseman, May 19 2020
First differences of A030303 = positions of bits 1 in the concatenation A030190 (= A030302) of numbers written in binary (A007088). - Indices of record values (= first occurrence of n) are given by A005183: a(A005183(n)) = n, cf. FORMULA for more. - M. F. Hasler, Oct 12 2020
The geometric mean approaches the Somos constant (A112302). - Jwalin Bhatt, Feb 10 2025

			A057335 begins 1 2 4 6 8 12 18 30 16 24 36 ... so we can write
  1 2 1 3 2 1 1 4 3 2 2 1 1 1 1 ...
  . . 1 . 1 2 1 . 1 2 1 3 2 1 1 ...
  . . . . . . 1 . . . 1 . 1 2 1 ...
  . . . . . . . . . . . . . . 1 ...
and the columns here gives the rows of the triangle, which begins
  1
  2; 1 1
  3; 2 1; 1 2; 1 1 1
  4; 3 1; 2 2; 2 1 1; 1 3; 1 2 1; 1 1 2; 1 1 1 1
  ...
From _Omar E. Pol_, Sep 03 2013: (Start)
Illustration of initial terms:
  -----------------------------------
  n  j       Diagram   Composition j
  -----------------------------------
  .               _
  1  1           |_|   1;
  .             _ _
  2  1         |  _|   2,
  2  2         |_|_|   1, 1;
  .           _ _ _
  3  1       |    _|   3,
  3  2       |  _|_|   2, 1,
  3  3       | |  _|   1, 2,
  3  4       |_|_|_|   1, 1, 1;
  .         _ _ _ _
  4  1     |      _|   4,
  4  2     |    _|_|   3, 1,
  4  3     |   |  _|   2, 2,
  4  4     |  _|_|_|   2, 1, 1,
  4  5     | |    _|   1, 3,
  4  6     | |  _|_|   1, 2, 1,
  4  7     | | |  _|   1, 1, 2,
  4  8     |_|_|_|_|   1, 1, 1, 1;
(End)

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..5120 (through compositions of 10)
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].
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Lists of compositions of integers: this sequence (reverse lexicographic order; minus one gives A108730), A228351 (reverse colexicographic order - every composition is reversed; minus one gives A163510), A228369 (lexicographic), A228525 (colexicographic), A124734 (length, then lexicographic; minus one gives A124735), A296774 (length, then reverse lexicographic), A337243 (length, then colexicographic), A337259 (length, then reverse colexicographic), A296773 (decreasing length, then lexicographic), A296772 (decreasing length, then reverse lexicographic), A337260 (decreasing length, then colexicographic), A108244 (decreasing length, then reverse colexicographic), also A101211 and A227736 (run lengths of bits).
Cf. row length and row sums for different splittings into rows: A000120, A070939, A001792, A001788.
Cf. A228531, A096903, A065120, A057335, A055932, A005811, A261300, A007088.
Cf. lists of partitions of integers, or multisets of integers: A026791 and crosserfs therein, A112798 and crossrefs therein.
See link for additional crossrefs pertaining to standard compositions.
A related ranking of finite sets is A048793/A272020.
Cf. A005183, A030190, A030302, A030303, A035327, A036036, A080576, A080577, A106356, A112302, A238279, A333219.

a066099 = (!!) a066099_list
a066099_list = concat a066099_tabf
a066099_tabf = map a066099_row [1..]
a066099_row n = reverse $ a228351_row n
-- (each composition as a row)
-- Peter Kagey, Aug 25 2016

Table[FactorInteger[Apply[Times, Map[Prime, Accumulate @ IntegerDigits[n, 2]]]][[All, -1]], {n, 41}] // Flatten (* Michael De Vlieger, Jul 11 2017 *)
stc[n_] := Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]] // Reverse;
Table[stc[n], {n, 0, 20}] // Flatten (* Gus Wiseman, May 19 2020 *)
Table[Reverse @ LexicographicSort @ Flatten[Permutations /@ Partitions[n], 1], {n, 10}] // Flatten (* Eric W. Weisstein, Jun 26 2023 *)

arow(n) = {local(v=vector(n),j=0,k=0);
   while(n>0,k++; if(n%2==1,v[j++]=k;k=0);n\=2);
   vector(j,i,v[j-i+1])} \\ returns empty for n=0. - Franklin T. Adams-Watters, Apr 02 2014

from itertools import islice
from itertools import accumulate, count, groupby, islice
def A066099_gen():
    for i in count(1):
        yield [len(list(g)) for _,g in groupby(accumulate(int(b) for b in bin(i)[2:]))]
A066099 = list(islice(A066099_gen(), 120))  # Jwalin Bhatt, Feb 28 2025

def a_row(n): return list(reversed(Compositions(n)))
flatten([a_row(n) for n in range(1,6)]) # Peter Luschny, May 19 2018

From M. F. Hasler, Oct 12 2020: (Start)
a(n) = A030303(n+1) - A030303(n).
a(A005183(n)) = n; a(A005183(n)+1) = n-1 (n>1); a(A005183(n)+2) = 1. (End)

Edited with additional terms by Franklin T. Adams-Watters, Nov 06 2006

0th row removed by Andrey Zabolotskiy, May 19 2018

A135010 Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of juxtaposed lexicographically ordered partitions of n that do not contain 1 as a part.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 17 2007, Mar 21 2008

This is the original sequence of a large number of sequences connected with the section model of partitions.
Here "the n-th section of the set of partitions of any integer greater than or equal to n" (hence "the last section of the set of partitions of n") is defined to be the set formed by all parts that occur as a result of taking all partitions of n and then removing all parts of the partitions of n-1. For integers greater than 1 the structure of a section has two main areas: the head and tail. The head is formed by the partitions of n that do not contain 1 as a part. The tail is formed by A000041(n-1) partitions of 1. The set of partitions of n contains the sets of partitions of the previous numbers. The section model of partitions has several versions according with the ordering of the partitions or with the representation of the sections. In this sequence we use the ordering of A026791.
The section model of partitions can be interpreted as a table of partitions. See also A138121. - Omar E. Pol, Nov 18 2009
It appears that the versions of the model show an overlapping of sections and subsections of the numbers congruent to k mod m into parts >= m. For example:
First generation (the main table):
Table 1.0: Partitions of integers congruent to 0 mod 1 into parts >= 1.
Second generation:
Table 2.0: Partitions of integers congruent to 0 mod 2 into parts >= 2.
Table 2.1: Partitions of integers congruent to 1 mod 2 into parts >= 2.
Third generation:
Table 3.0: Partitions of integers congruent to 0 mod 3 into parts >= 3.
Table 3.1: Partitions of integers congruent to 1 mod 3 into parts >= 3.
Table 3.2: Partitions of integers congruent to 2 mod 3 into parts >= 3.
And so on.
Conjecture:
Let j and n be integers congruent to k mod m such that 0 <= k < m <= j < n. Let h=(n-j)/m. Consider only all partitions of n into parts >= m. Then remove every partition in which the parts of size m appears a number of times < h. Then remove h parts of size m in every partition. The rest are the partitions of j into parts >= m. (Note that in the section model, h is the number of sections or subsections removed), (Omar E. Pol, Dec 05 2010, Dec 06 2010).
Starting from the first row of triangle, it appears that the total numbers of parts of size k in k successive rows give the sequence A000041 (see A182703). - Omar E. Pol, Feb 22 2012
The last section of n contains A187219(n) regions (see A206437). - Omar E. Pol, Nov 04 2012

			Triangle begins:
  [1];
  [1],[2];
  [1],[1],[3];
  [1],[1],[1],[2,2],[4];
  [1],[1],[1],[1],[1],[2,3],[5];
  [1],[1],[1],[1],[1],[1],[1],[2,2,2],[2,4],[3,3],[6];
  ...
From _Omar E. Pol_, Sep 03 2013: (Start)
Illustration of initial terms (n = 1..6). The table shows the six sections of the set of partitions of 6 in three ways. Note that before the dissection, the set of partitions was in the ordering mentioned in A026791. More generally, the six sections of the set of partitions of 6 also can be interpreted as the first six sections of the set of partitions of any integer >= 6.
---------------------------------------------------------
n  j          Diagram          Parts           Parts
---------------------------------------------------------
.                   _
1  1               |_|                1;              1;
.                 _
2  1             | |_               1,              1,
2  2             |_ _|              2;                2;
.               _
3  1           | |                1,              1,
3  2           | |_ _             1,                1,
3  3           |_ _ _|            3;                  3;
.             _
4  1         | |                1,              1,
4  2         | |                1,                1,
4  3         | |_ _ _           1,                  1,
4  4         |   |_ _|          2,2,                2,2,
4  5         |_ _ _ _|          4;                    4;
.           _
5  1       | |                1,              1,
5  2       | |                1,                1,
5  3       | |                1,                  1,
5  4       | |                1,                  1,
5  5       | |_ _ _ _         1,                    1,
5  6       |   |_ _ _|        2,3,                  2,3,
5  7       |_ _ _ _ _|        5;                      5;
.         _
6  1     | |                1,              1,
6  2     | |                1,                1,
6  3     | |                1,                  1,
6  4     | |                1,                  1,
6  5     | |                1,                    1,
6  6     | |                1,                    1,
6  7     | |_ _ _ _ _       1,                      1,
6  8     |   |   |_ _|      2,2,2,                2,2,2,
6  9     |   |_ _ _ _|      2,4,                    2,4,
6  10    |     |_ _ _|      3,3,                    3,3,
6  11    |_ _ _ _ _ _|      6;                        6;
...
(End)

Alois P. Heinz, Rows n = 1..23, flattened
Omar E. Pol, Illustration of the section model of partitions
Omar E. Pol, Illustration of the section model of partitions (2D view)
Omar E. Pol, Illustration of the section model of partitions (3D view)
Robert Price, Mathematica program to generate "Illustration of the section model of partitions"
Robert Price, Mathematica program to generate "Illustration of the section model of partitions (2D view)"

Row n has length A138137(n).
Row sums give A138879.
Right border gives A000027.
Cf. A000041, A026791, A138121, A141285, A182703, A187219, A193870, A194446, A206437, A207031, A207383, A207379, A211009.

with(combinat):
T:= proc(m) local b, ll;
      b:= proc(n, i, l)
            if n=0 then ll:=ll, l[]
          else seq(b(n-j, j, [l[], j]), j=i..n)
            fi
          end;
      ll:= NULL; b(m, 2, []); [1$numbpart(m-1)][], ll
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Feb 19 2012

less[run1_, run2_] := (lg1 = run1 // Length; lg2 = run2 // Length; lg = Max[lg1, lg2]; r1 = If[lg1 == lg, run1, PadRight[run1, lg, 0]]; r2 = If[lg2 == lg, run2, PadRight[run2, lg, 0]]; Order[r1, r2] != -1); row[n_] := Join[ Array[1 &, {PartitionsP[n - 1]}], Sort[ Reverse /@ Select[ IntegerPartitions[n], FreeQ[#, 1] &], less] ] // Flatten; Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Jan 14 2013 *)
Table[Reverse@ConstantArray[{1}, PartitionsP[n - 1]]~Join~
DeleteCases[Sort@PadRight[Reverse/@Cases[IntegerPartitions[n], x_ /; Last[x] != 1]], x_ /; x == 0, 2], {n, 1, 9}] // Flatten (* Robert Price, May 12 2020 *)

A080577 Triangle in which n-th row lists all partitions of n, in graded reverse lexicographic ordering.

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

Offset: 1

N. J. A. Sloane, Mar 23 2003

This is the "Mathematica" ordering of the partitions, referenced in numerous other sequences. The partitions of each integer are in reverse order of the conjugates of the partitions in Abramowitz and Stegun order (A036036). They are in the reverse of the order of the partitions in Maple order (A080576). - Franklin T. Adams-Watters, Oct 18 2006
The graded reverse lexicographic ordering of the partitions is often referred to as the "Canonical" ordering of the partitions. - Daniel Forgues, Jan 21 2011
Also the "MAGMA" ordering of the partitions. - Jason Kimberley, Oct 28 2011
Also an intuitive ordering described but not formalized in [Hardy and Wright] the first four editions of which precede [Abramowitz and Stegun]. - L. Edson Jeffery, Aug 03 2013
Also the "Sage" ordering of the partitions. - Peter Luschny, Aug 12 2013
While this is the order used for the constructive function "IntegerPartitions", it is different from Mathematica's canonical ordering of finite expressions, the latter giving A036036 if parts of partitions are read in reversed (weakly increasing) order, or A334301 if in the usual (weakly decreasing) order. - 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 colexicographic ordering A036037. The first row which differs is the sixth one, which reads ((6), (5,1), (4,2), (4,1,1), (3,3), (3,2,1), (3,1,1,1), (2,2,2), (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)    (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)
The triangle with partitions shown as Heinz numbers (A129129) begins:
   1
   2
   3   4
   5   6   8
   7  10   9  12  16
  11  14  15  20  18  24  32
  13  22  21  28  25  30  40  27  36  48  64
  17  26  33  44  35  42  56  50  45  60  80  54  72  96 128
(End)

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, Fifth edition, 1979, p. 273.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 287.

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, p. 831.
OEIS Wiki, Orderings of partitions (a comparison).
Sergei Viznyuk, C Program
Wikiversity, Lexicographic and colexicographic order

See A080576 Maple (graded reflected lexicographic) ordering.
See A036036 for the Hindenburg (graded reflected colexicographic) ordering (listed in the Abramowitz and Stegun Handbook).
See A036037 for graded colexicographic ordering.
See A228100 for the Fenner-Loizou (binary tree) ordering.
Differs from A036037 at a(48).
See A322761 for a compressed version.
Lexicographically ordered reversed partitions are A026791.
Reverse-colexicographically ordered partitions are A026792.
Compositions under this ordering are A066099.
Distinct parts of these partitions are counted by A115623.
Taking Heinz numbers gives A129129.
Lexicographically ordered partitions are A193073.
Colexicographically ordered partitions are A211992.
Reading partitions in reverse (weakly increasing) order gives A228531.
Lengths of these partitions are A238966.
Sorting partitions by Heinz number gives A296150.
The maxima of these partitions are A331581.
The length-sensitive version is A334439.
Cf. A000041, A048793, A063008, A185974, A334301, A334434, A334436, A334438.

&cat[&cat Partitions(n):n in[1..7]]; // Jason Kimberley, Oct 28 2011

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-> x[], b(n$2))[]:
seq(T(n), n=1..8);  # Alois P. Heinz, Jan 29 2020

<Jean-François Alcover, Dec 10 2012 *)
revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
Join@@Table[Sort[IntegerPartitions[n],revlexsort],{n,0,8}] (* Gus Wiseman, May 08 2020 *)

A080577_row(n)={vecsort(apply(t->Vecrev(t),partitions(n)),,4)} \\ M. F. Hasler, Jan 21 2020

L = []
for n in range(8): L += list(Partitions(n))
flatten(L)   # Peter Luschny, Aug 12 2013

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.

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, 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

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

A036043 Irregular triangle read by rows: row n (n >= 0) gives number of parts in all partitions of n (in Abramowitz and Stegun 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, 2, 3, 3, 3, 4, 4, 5, 6, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 6, 6, 7, 8, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 8, 9

Offset: 0

N. J. A. Sloane

The sequence of row lengths of this array is p(n) = A000041(n) (partition numbers).
The sequence of row sums is A006128(n).
The number of times k appears in row n is A008284(n,k). - Franklin T. Adams-Watters, Jan 12 2006
The next level (row) gets created from each node by adding one or two more nodes. If a single node is added, its value is one more than the value of its parent. If two nodes are added, the first is equal in value to the parent and the value of the second is one more than the value of the parent. See A128628. - Alford Arnold, Mar 27 2007
The 1's in the (flattened) sequence mark the start of a new row, the value that precedes the 1 equals the row number minus one. (I.e., the 1 preceded by a 0 is the start of row 1, the 1 preceded by a 6 is the start of row 7, etc.) - M. F. Hasler, Jun 06 2018
Also the maximum part in the n-th partition in graded lexicographic order (sum/lex, A193073). - Gus Wiseman, May 24 2020

			0;
1;
1, 2;
1, 2, 3;
1, 2, 2, 3, 4;
1, 2, 2, 3, 3, 4, 5;
1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6;
1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7;

Abramowitz and Stegun, Handbook, p. 831, column labeled "m".

T. D. Noe, Rows n = 0..25 of irregular triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [scanned copy], p. 831.
Kevin Brown, Generalized Birthday Problem (N Items in M Bins), 1994-2010.
Wolfdieter Lang, Rows n = 1 ..20.
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Cf. A036037, A036038, A036039, A036040, A036042.
Row lengths are A000041.
Partition lengths of A036036 and A334301.
The version not sorted by length is A049085.
The generalization to compositions is A124736.
The Heinz number of the same partition is A334433.
The number of distinct elements in the same partition is A334440.
The maximum part of the same partition is A334441.
Lexicographically ordered reversed partitions are A026791.
Lexicographically ordered partitions are A193073.
Cf. A103921, A115623, A124734, A185974, A296150, A334435, A334438, A334439.

with(combinat): nmax:=9: for n from 1 to nmax do y(n):=numbpart(n): P(n):=sort(partition(n)): for k from 1 to y(n) do B(k) := P(n)[k] od: for k from 1 to y(n) do s:=0: j:=0: while sJohannes W. Meijer, Jun 21 2010, revised Nov 29 2012
# alternative implementation based on A119441 by R. J. Mathar, Jul 12 2013
A036043 := proc(n,k)
    local pi;
    pi := ASPrts(n)[k] ;
    nops(pi) ;
end proc:
for n from 1 to 10 do
    for k from 1 to A000041(n) do
        printf("%d,",A036043(n,k)) ;
    end do:
    printf("\n") ;
end do:

Table[Length/@Sort[IntegerPartitions[n]],{n,0,30}] (* Gus Wiseman, May 22 2020 *)

A036043(n,k)=#partitions(n)[k] \\ M. F. Hasler, Jun 06 2018

def A036043_row(n):
    return [len(p) for k in (0..n) for p in Partitions(n, length=k)]
for n in (0..10): print(A036043_row(n)) # Peter Luschny, Nov 02 2019

a(n) = A001222(A334433(n)). - Gus Wiseman, May 22 2020

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001
a(0) inserted by Franklin T. Adams-Watters, Jun 24 2014
Incorrect formula deleted by M. F. Hasler, Jun 06 2018

cp's OEIS Frontend

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

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A318271 The optimum crossing time for the Bridge and Torch problem, given that the crossing times for the group's members are given by the n-th partition in A026791.

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A066099 Triangle read by rows, in which row n lists the compositions of n in reverse lexicographic order.

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A135010 Triangle read by rows in which row n lists A000041(n-1) 1's followed by the list of juxtaposed lexicographically ordered partitions of n that do not contain 1 as a part.

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A080577 Triangle in which n-th row lists all partitions of n, in graded reverse lexicographic ordering.

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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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