A128628 -id:A128628 - cp's OEIS frontend

A308355 Limiting row sequence of the array A128628.

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

Offset: 1

Clark Kimberling, May 24 2019

Conjecture:  The length of maximal initial segment of this sequence that is identical to row n of A128628 is A025065(n+1), for n >= 1.
Beginning with the 2nd term, the sequence is a concatenation of segments that begin with 2:
2
2, 3
2, 3, 4
2, 3, 3, 4, 5
2, 3, 3, 4, 4, 5, 6
2, 3, 3, 4, 3, 4, 5, 4, 5, 6, 7
2, 3, 3, 4, 3, 4, 5, 4, 4, 5, 6, 5, 6, 7, 8

			Successive rows of A128628 (as in Jason Kimberley's comment: in row n, the k-th term is the number of parts in the k-th partition of n, assuming the parts of each partition are in nonincreasing order):
  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

Clark Kimberling, Table of n, a(n) for n = 1..2000

Cf. A000041, A025065, A128628.

Take[Map[Length, IntegerPartitions[50]], 1000]

A129127 An irregular table of node numbers (in order of appearance) which generate two children in the tree described by A128628 and A129129.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 14, 15, 18, 13, 22, 21, 30, 17, 26, 33, 35, 42, 50, 54, 19, 34, 39, 55, 66, 70, 75, 90

Offset: 1

Alford Arnold, Apr 01 2007

Shape sequence for A129127 is A002865 since the odd numbers can be mapped to cyclic partitions. The sequence contains the same values as A129128 which is sorted by the associated odd numbers.

Cf. A002865, A128628, A129128, A129129.

A025065 Number of palindromic partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 19, 19, 30, 30, 45, 45, 67, 67, 97, 97, 139, 139, 195, 195, 272, 272, 373, 373, 508, 508, 684, 684, 915, 915, 1212, 1212, 1597, 1597, 2087, 2087, 2714, 2714, 3506, 3506, 4508, 4508, 5763, 5763, 7338, 7338, 9296, 9296, 11732, 11732, 14742, 14742, 18460, 18460, 23025, 23025, 28629, 28629

Offset: 0

Clark Kimberling

nonn

That is, the number of partitions of n into parts which can be listed in palindromic order.
Alternatively, number of partitions of n into parts from the set {1,2,4,6,8,10,12,...}. - T. D. Noe, Aug 05 2005
Also, partial sums of A035363.
Also number of partitions of n with at most one part occurring an odd number of times. - Reinhard Zumkeller, Dec 18 2013
The first Mathematica program computes terms of A025065; the second computes the k palindromic partitions of user-chosen n. - Clark Kimberling, Jan 20 2014
a(n) is the number of partitions p of n+1 such that 2*max(p) > n+1. - Clark Kimberling, Apr 20 2014.
From Gus Wiseman, Nov 28 2018: (Start)
Also the number of integer partitions of n + 2 that are the vertex-degrees of some hypertree. For example, the a(6) = 7 partitions of 8 that are the vertex-degrees of some hypertree, together with a realizing hypertree are:
     (41111): {{1,2},{1,3},{1,4},{1,5}}
     (32111): {{1,2},{1,3},{1,4},{2,5}}
     (22211): {{1,2},{1,3},{2,4},{3,5}}
    (311111): {{1,2},{1,3},{1,4,5,6}}
    (221111): {{1,2},{1,3},{2,4,5,6}}
   (2111111): {{1,2},{1,3,4,5,6,7}}
  (11111111): {{1,2,3,4,5,6,7,8}}
(End)
Conjecture: a(n) is the length of maximal initial segment of A308355(n-1) that is identical to row n of A128628, for n >= 2. - Clark Kimberling, May 24 2019
From Gus Wiseman, May 21 2021: (Start)
The Heinz numbers of palindromic partitions are given by A265640. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
Also the number of integer partitions of n with a part greater than or equal to n/2. This is equivalent to Clark Kimberling's final comment above. The Heinz numbers of these partitions are given by A344414. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)     (8)
       (11)  (21)  (22)   (32)   (33)    (43)    (44)
                   (31)   (41)   (42)    (52)    (53)
                   (211)  (311)  (51)    (61)    (62)
                                 (321)   (421)   (71)
                                 (411)   (511)   (422)
                                 (3111)  (4111)  (431)
                                                 (521)
                                                 (611)
                                                 (4211)
                                                 (5111)
                                                 (41111)
Also the number of integer partitions of n with at least n/2 parts. The Heinz numbers of these partitions are given by A344296. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (2)   (21)   (22)    (221)    (222)     (2221)     (2222)
       (11)  (111)  (31)    (311)    (321)     (3211)     (3221)
                    (211)   (2111)   (411)     (4111)     (3311)
                    (1111)  (11111)  (2211)    (22111)    (4211)
                                     (3111)    (31111)    (5111)
                                     (21111)   (211111)   (22211)
                                     (111111)  (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

			The partitions for the first few values of n are as follows:
n: partitions .......................... number
1: 1 ................................... 1
2: 2 11 ................................ 2
3: 3 111 ............................... 2
4: 4 22 121 1111 ....................... 4
5: 5 131 212 11111 ..................... 4
6: 6 141 33 222 1221 11211 111111 ...... 7
7: 7 151 313 11311 232 21112 1111111 ... 7
From _Reinhard Zumkeller_, Jan 23 2010: (Start)
Partitions into 1,2,4,6,... for the first values of n:
1: 1 ....................................... 1
2: 2 11 .................................... 2
3: 21 111 .................................. 2
4: 4 22 211 1111 ........................... 4
5: 41 221 2111 11111 ....................... 4
6: 6 42 4211 222 2211 21111 111111.......... 7
7: 61 421 42111 2221 22111 211111 1111111 .. 7. (End)

David A. Corneth, Table of n, a(n) for n = 0..10000 (first 101 terms from Reinhard Zumkeller that were corrected by _Georg Fischer_, Jan 20 2019)

Cf. A172033, A004277. - Reinhard Zumkeller, Jan 23 2010
Cf. A004526, A030019, A056503, A147878, A320921, A322136.
The bisections are both A000070.
The ordered version (palindromic compositions) is A016116.
The complement is counted by A233771 and A210249.
The case of palindromic prime signature is A242414.
Palindromic partitions are ranked by A265640, with complement A229153.
The case of palindromic plane trees is A319436.
The multiplicative version (palindromic factorizations) is A344417.
A000569 counts graphical partitions.
A027187 counts partitions of even length, ranked by A028260.
A035363 counts partitions into even parts, ranked by A066207.
A058696 counts partitions of even numbers, ranked by A300061.
A110618 counts partitions with length <= half sum, ranked by A344291.
Cf. A000041, A067538, A143773, A209816, A338914, A338915, A340387, A344296, A344414, A344415, A344416.

a025065 = p (1:[2,4..]) where
   p [] _ = 0
   p _  0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 12 2011

import Data.List (group)
a025065 = length . filter (<= 1) .
                   map (sum . map ((`mod` 2) . length) . group) . ps 1
   where ps x 0 = [[]]
         ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Dec 18 2013

Map[Length[Select[IntegerPartitions[#], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &]] &, Range[40]] (* Peter J. C. Moses, Jan 20 2014 *)
n = 8; Select[IntegerPartitions[n], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &] (* Peter J. C. Moses, Jan 20 2014 *)
CoefficientList[Series[1/((1 - x) Product[1 - x^(2 n), {n, 1, 50}]), {x, 0, 60}], x] (* Clark Kimberling, Mar 14 2014 *)

N=66; q='q+O('q^N); Vec( 1/((1-q)*eta(q^2)) ) \\ Joerg Arndt, Mar 11 2014

a(n) = A000070(A004526(n)). - Reinhard Zumkeller, Jan 23 2010
G.f.: 1/((1-q)*prod(n>=1, 1-q^(2*n))). [Joerg Arndt, Mar 11 2014]
a(2*k+2) = a(2*k) + A000041(k + 1). - David A. Corneth, May 29 2021
a(n) ~ exp(Pi*sqrt(n/3)) / (2*Pi*sqrt(n)). - Vaclav Kotesovec, Nov 16 2021

Edited by N. J. A. Sloane, Dec 29 2007

Prepended a(0)=1, added more terms, Joerg Arndt, Mar 11 2014

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

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)

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

A333485 Heinz numbers of all integer partitions sorted first by sum, then by decreasing length, and finally lexicographically. A code for the Fenner-Loizou tree A228100.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 11 2020

A permutation of the positive integers.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), which gives a bijective correspondence between positive integers and integer partitions.
As a triangle with row lengths A000041, the sequence starts {{1},{2},{4,3},{8,6,5},...}, so offset is 0.

			The sequence of terms together with their prime indices begins:
    1: {}              11: {5}                 56: {1,1,1,4}
    2: {1}             64: {1,1,1,1,1,1}       45: {2,2,3}
    4: {1,1}           48: {1,1,1,1,2}         50: {1,3,3}
    3: {2}             36: {1,1,2,2}           42: {1,2,4}
    8: {1,1,1}         40: {1,1,1,3}           44: {1,1,5}
    6: {1,2}           27: {2,2,2}             35: {3,4}
    5: {3}             30: {1,2,3}             33: {2,5}
   16: {1,1,1,1}       28: {1,1,4}             26: {1,6}
   12: {1,1,2}         25: {3,3}               17: {7}
    9: {2,2}           21: {2,4}              256: {1,1,1,1,1,1,1,1}
   10: {1,3}           22: {1,5}              192: {1,1,1,1,1,1,2}
    7: {4}             13: {6}                144: {1,1,1,1,2,2}
   32: {1,1,1,1,1}    128: {1,1,1,1,1,1,1}    160: {1,1,1,1,1,3}
   24: {1,1,1,2}       96: {1,1,1,1,1,2}      108: {1,1,2,2,2}
   18: {1,2,2}         72: {1,1,1,2,2}        120: {1,1,1,2,3}
   20: {1,1,3}         80: {1,1,1,1,3}        112: {1,1,1,1,4}
   15: {2,3}           54: {1,2,2,2}           81: {2,2,2,2}
   14: {1,4}           60: {1,1,2,3}           90: {1,2,2,3}
The triangle begins:
    1
    2
    4   3
    8   6   5
   16  12   9  10   7
   32  24  18  20  15  14  11
   64  48  36  40  27  30  28  25  21  22  13
  128  96  72  80  54  60  56  45  50  42  44  35  33  26  17

Michael De Vlieger, Table of n, a(n) for n = 0..9295 (rows 0 <= n <= 25, flattened)
Michael De Vlieger, log-log plot of rows 0 <= n <= 30 of this sequence, highlighting 2^n in red and prime(n) in blue.
T. I. Fenner, G. Loizou: A binary tree representation and related algorithms for generating integer partitions. The Computer J. 23(4), 332-337 (1980)
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The constructive version is A228100.
Sorting by increasing length gives A334433.
The version with rows reversed is A334438.
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.
Graded Heinz numbers are A215366.
Sorting partitions by Heinz number gives A296150.
If the fine ordering is by Heinz number instead of lexicographic we get A333484.
Cf. A000041, A026791, A036036, A036037, A036043, A185974, A211992, A228351, A296773, A333483, A334301, A334439.

ralensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]>Length[c],OrderedQ[{f,c}]];
Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n],ralensort],{n,0,8}]

A001221(a(n)) = A115623(n).
A001222(a(n - 1)) = A331581(n).
A061395(a(n > 1)) = A128628(n).

Name extended by Peter Luschny, Dec 23 2020

A129128 List of nodes generating two branches in the tree defined in sequence A129129.

Original entry on oeis.org

2, 3, 5, 6, 7, 11, 10, 13, 17, 14, 19, 15, 18, 23, 29, 22, 21, 31, 26, 37, 41, 30, 43, 35, 34, 47, 33, 38, 53, 59, 42, 39, 61, 46, 67, 71, 50, 55, 73, 54, 79, 51, 58, 83, 65, 62, 57, 89, 66, 97, 101, 70, 103, 107, 74, 109, 69, 78, 85, 77, 82, 75, 113, 86, 127, 95, 90, 131, 137

Offset: 1

Alford Arnold, Mar 31 2007

The odd numbers begin 1 3 5 7 9 11 ... therefore a(n) begins 2 3 5 6 7 ...

			The tree begins
  1
  2
  3 4
  5 6 8
  7 10 9 12 16
  11 14 15 20 18 24 32
  ...
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 adjusted 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 generating the odd number 135 is (2/3)*135 = 90.
therefore 90 is a member of A129128.

Cf. A020639, A129129.

A177740 Triangle read by rows in which row n lists the number of parts in each of those partitions of n that have every part >=3.

Original entry on oeis.org

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

Offset: 3

Jason Kimberley, May 13 2010

The ordering of the partitions is described in A176210.

			3: 3 (1)
4: 4 (1)
5: 5 (1)
6: 6 or 3,3 (1, 2)
7: 7 or 3,4 (1, 2)
8: 8, 3,5 or 4,4 (1, 2, 2)
so sequence begins 1, 1, 1, 1,2, 1,2, ...

J. S. Kimberley, Rows 3..39 of A177740 triangle, flattened.

The length of the rows is given by A008483.

Cf. A128628 for unrestricted partitions.

[ #part: part in RestrictedPartitions(n,{3..n}), n in [0..16]];

Edited (including b-file) by N. J. A. Sloane, Oct 05 2010

A279038 Triangle of multinomial coefficients read by rows (ordered by decreasing size of the greatest part).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 6, 8, 3, 6, 1, 24, 30, 20, 20, 15, 10, 1, 120, 144, 90, 90, 40, 120, 40, 15, 45, 15, 1, 720, 840, 504, 504, 420, 630, 210, 280, 210, 420, 70, 105, 105, 21, 1, 5040, 5760, 3360, 3360, 2688, 4032, 1344, 1260, 3360, 1260, 2520, 420, 1120, 1120, 1680, 1120, 112, 105, 420, 210, 28, 1

Offset: 0

David W. Wilson and Olivier Gérard, Dec 04 2016

The ordering of integer partitions used in this version is also called:
- canonical ordering
- graded reverse lexicographic ordering
- magma (software) ordering
by opposition to the ordering used by Abramowitz and Stegun.

			First rows are:
    1
    1
    1   1
    2   3   1
    6   8   3   6   1
   24  30  20  20  15   10   1
  120 144  90  90  40  120  40  15  45  15  1
  720 840 504 504 420  630 210 280 210 420 70 105 105 21 1
  ...

Alois P. Heinz, Rows n = 0..28, flattened

Cf. A000041 (number of partitions of n, length of each row).
Cf. A128628 (triangle of partition lengths)
Cf. A036039 (a different ordering), A102189 (row reversed version of A036039).
Row sums give A000142.

b:= proc(n, i) option remember; `if`(n=0, [1],
      `if`(i<1, [], [seq(map(x-> x*i^j*j!,
       b(n-i*j, i-1))[], j=[iquo(n, i)-t$t=0..n/i])]))
    end:
T:= n-> map(x-> n!/x, b(n$2))[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Dec 04 2016

Flatten[Table[
  Map[n!/Times @@ ((First[#]^Length[#]*Factorial[Length[#]]) & /@
        Split[#]) &, IntegerPartitions[n]], {n, 1, 8}]]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, {1},
     If[i < 1, {}, Flatten@Table[#*i^j*j!& /@
     b[n - i*j, i - 1], {j, Quotient[n, i] - Range[0, n/i]}]]];
T[n_] := n!/#& /@ b[n, n];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, Jun 01 2021, after Alois P. Heinz *)

One term for row n=0 prepended by Alois P. Heinz, Dec 04 2016

cp's OEIS Frontend

A308355 Limiting row sequence of the array A128628.

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A129127 An irregular table of node numbers (in order of appearance) which generate two children in the tree described by A128628 and A129129.

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A025065 Number of palindromic partitions of n.

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

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

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A333485 Heinz numbers of all integer partitions sorted first by sum, then by decreasing length, and finally lexicographically. A code for the Fenner-Loizou tree A228100.

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