A036042 -id:A036042 - cp's OEIS frontend

A238965 Duplicate of A036042.

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

Offset: 0

dead

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

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

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

A129129 An irregular triangular array of natural numbers read by rows, with shape sequence A000041(n) related to sequence A060850.

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, 28, 25, 30, 40, 27, 36, 48, 64, 17, 26, 33, 44, 35, 42, 56, 50, 45, 60, 80, 54, 72, 96, 128, 19, 34, 39, 52, 55, 66, 88, 49, 70, 63, 84, 112, 75, 100, 90, 120, 160, 81, 108, 144, 192, 256

Offset: 0

Alford Arnold, Mar 31 2007

The tree begins (at height n, n >= 0, nodes represent partitions of n)
0:  1
1:  2
2:  3  4
3:  5  6  8
4:  7 10  9 12 16
5: 11 14 15 20 18 24 32
...
and hence differs from A114622.
Ordering [graded reverse lexicographic order] of partitions (positive integer representation) of nonnegative integers, where part of size i [as summand] is mapped to i-th prime [as multiplicand], where the empty partition for 0 yields the empty product, i.e., 1. Permutation of positive integers, since bijection [1-1 and onto map] between the set of all partitions of nonnegative integers and positive integers. - Daniel Forgues, Aug 07 2018
These are all Heinz numbers of integer partitions in graded reverse-lexicographic order, where The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This is the so-called "Mathematica" order (sum/revlex) of partitions (A080577). Partitions in lexicographic order (sum/lex) are A193073, with Heinz numbers A334434. - Gus Wiseman, May 19 2020

			The array is a tree structure as described by A128628. If a node value has only one branch the value is twice that of its parent node. If it has two branches one is twice that of its parent node but the other is defined as indicated below:
(1) pick an odd number (e.g., 135)
(2) calculate its prime factorization (135 = 5*3*3*3)
(3) note the least prime factor (LPF(135) = 3)
(4) note the index of the LPF (index(3) = 2)
(5) subtract one from the index (2-1 = 1)
(6) calculate the prime associated with the value in step five (prime(1) = 2)
(7) The parent node of the odd number 135 is (2/3)*135 = 90 = A252461(135).
From _Daniel Forgues_, Aug 07 2018: (Start)
Partitions of 4 in graded reverse lexicographic order:
{4}: p_4 = 7;
{3,1}: p_3 * p_1 = 5 * 2 = 10;
{2,2}: p_2 * p_2 = 3^2 = 9;
{2,1,1}: p_2 * p_1 * p_1 = 3 * 2^2 = 12;
{1,1,1,1}: p_1 * p_1 * p_1 * p_1 = 2^4 = 16. (End)
From _Gus Wiseman_, May 19 2020: (Start)
The sequence together with the corresponding partitions begins:
    1: ()            24: (2,1,1,1)         35: (4,3)
    2: (1)           32: (1,1,1,1,1)       42: (4,2,1)
    3: (2)           13: (6)               56: (4,1,1,1)
    4: (1,1)         22: (5,1)             50: (3,3,1)
    5: (3)           21: (4,2)             45: (3,2,2)
    6: (2,1)         28: (4,1,1)           60: (3,2,1,1)
    8: (1,1,1)       25: (3,3)             80: (3,1,1,1,1)
    7: (4)           30: (3,2,1)           54: (2,2,2,1)
   10: (3,1)         40: (3,1,1,1)         72: (2,2,1,1,1)
    9: (2,2)         27: (2,2,2)           96: (2,1,1,1,1,1)
   12: (2,1,1)       36: (2,2,1,1)        128: (1,1,1,1,1,1,1)
   16: (1,1,1,1)     48: (2,1,1,1,1)       19: (8)
   11: (5)           64: (1,1,1,1,1,1)     34: (7,1)
   14: (4,1)         17: (7)               39: (6,2)
   15: (3,2)         26: (6,1)             52: (6,1,1)
   20: (3,1,1)       33: (5,2)             55: (5,3)
   18: (2,2,1)       44: (5,1,1)           66: (5,2,1)
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..11731 (rows 0 <= n <= 26).
OEIS Wiki, Partitions#Orderings of partitions.
Wikiversity, Lexicographic and colexicographic order.

Cf. A080577 (the partitions), A252461, A114622, A128628, A215366 (sorted rows).
Row lengths are A000041.
Compositions under the same order are A066099.
The opposite version (sum/lex) is A334434.
The length-sensitive version (sum/length/revlex) is A334438.
The version for reversed (weakly increasing) partitions is A334436.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
Sum of prime indices is A056239.
Sorting reversed partitions by Heinz number gives A112798.
Partitions in lexicographic order are A193073.
Sorting partitions by Heinz number gives A296150.
Cf. A036037, A185974, A211992, A334301, A334433, A334435, A334437, A334439.

b:= (n, i)-> `if`(n=0 or i=1, [2^n], [map(x-> x*ithprime(i),
                b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> b(n$2)[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 14 2020

Array[Times @@ # & /@ Prime@ IntegerPartitions@ # &, 9, 0] // Flatten (* Michael De Vlieger, Aug 07 2018 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {2^n}, Join[(# Prime[i]&) /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
T[n_] := b[n, n];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

From Gus Wiseman, May 19 2020: (Start)
A001222(a(n)) = A238966(n).
A001221(a(n)) = A115623(n).
A056239(a(n)) = A036042(n).
A061395(a(n)) = A331581(n).
(End)

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 *)

A049085 Irregular table T(n,k) = maximal part of the k-th partition of n, when listed in Abramowitz-Stegun order (as in A036043).

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 1, 4, 3, 2, 2, 1, 5, 4, 3, 3, 2, 2, 1, 6, 5, 4, 3, 4, 3, 2, 3, 2, 2, 1, 7, 6, 5, 4, 5, 4, 3, 3, 4, 3, 2, 3, 2, 2, 1, 8, 7, 6, 5, 4, 6, 5, 4, 4, 3, 5, 4, 3, 3, 2, 4, 3, 2, 3, 2, 2, 1, 9, 8, 7, 6, 5, 7, 6, 5, 4, 5, 4, 3, 6, 5, 4, 4, 3, 3, 5, 4, 3, 3, 2, 4, 3, 2, 3, 2, 2, 1, 10, 9, 8, 7, 6, 5, 8, 7, 6

Offset: 0

Alford Arnold

a(0) = 0 by convention. - Franklin T. Adams-Watters, Jun 24 2014
Like A036043 this is important for calculating sequences defined over the numeric partitions, cf. A000041. For example, the triangular array A019575 can be calculated using A036042 and this sequence.
The row sums are A006128. - Johannes W. Meijer, Jun 21 2010
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 A334441. - Gus Wiseman, May 21 2020

			Rows:
  [0];
  [1];
  [2,1];
  [3,2,1];
  [4,3,2,2,1];
  [5,4,3,3,2,2,1];
  ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.

Alois P. Heinz, Rows n = 0..26, 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].
Wolfdieter Lang, First 15 rows.
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
Row sums are A006128.
The length of the partition is A036043.
The number of distinct elements of the partition is A103921.
The Heinz number of the partition is A185974.
The version ignoring length is A194546.
The version for non-reversed partitions is A334441.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun order are A036036.
Reverse-lexicographically ordered partitions are A080577.
Partitions in Abramowitz-Stegun order are A334301.
Cf. A001221, A036037, A036042, A115623, A124734, A193073, A334302, A334433, A334438, A334439, A334440.

with(combinat):
nmax:=9:
for n from 1 to nmax do
   y(n):=numbpart(n):
   P(n):=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

Table[If[n==0,{0},Max/@Sort[Reverse/@IntegerPartitions[n]]],{n,0,8}] (* Gus Wiseman, May 21 2020 *)

A049085(n,k)=if(n,partitions(n)[k][1],0) \\ M. F. Hasler, Jun 06 2018

More terms from Wolfdieter Lang, Apr 28 2005

a(0) inserted by Franklin T. Adams-Watters, Jun 24 2014

A227183 a(n) is the sum of parts of the unique unordered partition encoded in the run lengths of the binary expansion of n; row sums of A227739 for n >= 1.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 3, 3, 6, 5, 4, 6, 5, 4, 4, 4, 8, 7, 6, 8, 8, 5, 7, 9, 7, 6, 5, 7, 6, 5, 5, 5, 10, 9, 8, 10, 10, 7, 9, 11, 12, 9, 6, 10, 11, 8, 10, 12, 9, 8, 7, 9, 9, 6, 8, 10, 8, 7, 6, 8, 7, 6, 6, 6, 12, 11, 10, 12, 12, 9, 11, 13, 14, 11, 8, 12, 13, 10, 12, 14

Offset: 0

Antti Karttunen, Jul 05 2013

Like A129594 this sequence utilizes the fact that compositions (i.e., ordered partitions) can be bijectively mapped to (unordered) partitions by taking the partial sums of the list of composants after one has been subtracted from each except the first one. Compositions in turn are mapped to nonnegative integers via the runlength encoding, where the lengths of maximum runs of 0's or 1's in binary representation of n give the composants. See the OEIS Wiki page and the example below.

Each n occurs A000041(n) times in total and occurs for the first time at A227368(n) and for the last time at position A000225(n). See further comments and conjectures at A227368 and A227370.

			19 has binary expansion "10011", thus the maximal runs of identical bits (scanned from right to left) are [2,2,1]. We subtract one from each after the first one, to get [2,1,0] and then form their partial sums as [2,2+1,2+1+0], which thus maps to unordered partition {2+3+3} which adds to 8. Thus a(19)=8.

Antti Karttunen, Table of n, a(n) for n = 0..8192
A. Karttunen et al., Run-length encoding, OEIS Wiki.
Index entries for sequences related to partitions

Row sums of A227189 and A227739. Cf. A227184 (corresponding products), A227185, A227189, A227192, A129594, A226062, A227368.

Analogous sum sequences computed for other encoding schemes of unordered partitions: A036042, A056239, A161511, A243503. Cf. also A229119, A003188, A075157, A243353 (associated permutations mapping between these schemes).

Table[Function[b, Total@ Accumulate@ Prepend[If[Length@ b > 1, Rest[b] - 1, {}], First@ b] - Boole[n == 0]]@ Map[Length, Split@ Reverse@ IntegerDigits[n, 2]], {n, 0, 79}] // Flatten (* Michael De Vlieger, May 09 2017 *)

def A227183(n):
  '''Sum of parts of the unique unordered partition encoded in the run lengths of the binary expansion of n.'''
  s = 0
  b = n%2
  i = 1
  while (n != 0):
    n >>= 1
    if ((n%2) == b): # Staying in the same run of bits?
      i += 1
    else: # The run changes.
      b = n%2
      s += i
  return(s)

a(n) = Sum_{i=0..A005811(n)-1} A227189(n,i). [The defining formula]
Equivalently, for n>=1, a(n) = Sum_{i=(A173318(n-1)+1)..A173318(n)} A227739(i).
a(n) = A227192(n) - A000217(A005811(n)-1).
Other identities:
a(A129594(n)) = a(n). [This follows from the fact that conjugating a partition doesn't change its total sum]
a(A226062(n)) = a(n). [Which is also true for the "Bulgarian operation"]
From Antti Karttunen, Mar 08 2015: (Start)
Can be also obtained by mapping with an appropriate permutation from the sequences giving sizes of each partition (i.e., sum of their parts) computed for other enumerations similar to A227739:
a(n) = A036042(A229119(n)).
a(n) = A161511(A003188(n)).
a(n) = A056239(A243353(n)).
a(n) = A243503(1+A075157(n)).
(End)

A161924 Permutation of natural numbers: sequence A126441 without zeros.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 6, 11, 15, 16, 17, 10, 19, 13, 23, 31, 32, 33, 18, 35, 12, 21, 14, 39, 27, 47, 63, 64, 65, 34, 67, 20, 37, 22, 71, 25, 43, 29, 79, 55, 95, 127, 128, 129, 66, 131, 36, 69, 38, 135, 24, 41, 26, 75, 45, 30, 143, 51, 87, 59, 159, 111, 191, 255, 256

Offset: 1

Alford Arnold, Jun 23 2009

Values appear in the order determined by A004760(n+1)and A062383(n).

The graph of this sequence looks very elegant.

			The table begins:
1.2.4..8.16.32.64.128.256.512.1024
..3.5..9.17.33.65.129.257.513.1025
.......6.10.18.34..66.130.258..514
....7.11.19.35.67.131.259.515.1027
............12.20..36..68.132..260
.........13.21.37..69.133.261..517
............14.22..38..70.134..262
......15.23.39.71.135.263.519.1031
...................24..40..72..136
...............25..41..73.137..265
...................26..42..74..138
............27.43..75.139.267..523
.......................28..44...76
...............29..45..77.141..269
...................30..46..78..142
.........31.47.79.143.271.527.1039
...........................48...80
.......................49..81..145
...........................50...82
...................51..83.147..275
This can be viewed as an irregular table, where row r (>= 1) has A000041(r) elements, that is, as 1; 2,3; 4,5,7; 8,9,6,11,15; 16,17,10,19,13,23,31; etc. A125106 illustrates how each number is mapped to a partition.

Inverse: A166276. a(n) = A126441(A166274(n)). See A161919 for the version with each row sorted into ascending order.

A161511(a(n)) = A036042(n).

columns = 9; row[n_] := n - 2^Floor[Log2[n]]; col[0] = 0; col[n_] := If[EvenQ[n], col[n/2] + DigitCount[n/2, 2, 1], col[(n - 1)/2] + 1]; Clear[T]; T[, ] = 0; Do[T[row[k], col[k]] = k, {k, 1, 2^columns}]; Table[DeleteCases[Table[T[n - 1, k], {n, 1, 2^(k - 1)}], 0], {k, 1, columns}] // Flatten (* Jean-François Alcover, Sep 09 2017 *)

Edited and extended by Antti Karttunen, Oct 12 2009

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

cp's OEIS Frontend

A238965 Duplicate of A036042.

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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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A036043 Irregular triangle read by rows: row n (n >= 0) gives number of parts in all partitions of n (in Abramowitz and Stegun order).

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

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A129129 An irregular triangular array of natural numbers read by rows, with shape sequence A000041(n) related to sequence A060850.

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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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A049085 Irregular table T(n,k) = maximal part of the k-th partition of n, when listed in Abramowitz-Stegun order (as in A036043).

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