A228351 -id:A228351 - cp's OEIS frontend

A334433 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally lexicographically.

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

Offset: 0

Gus Wiseman, Apr 30 2020

First differs from A334435 at a(75) = 99, A334435(75) = 98.
A permutation of the positive integers.
This is the Abramowitz-Stegun ordering of integer partitions when the parts are read in the usual (weakly decreasing) order. The case of reversed (weakly increasing) partitions is A185974.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.

			The sequence of terms together with their prime indices begins:
    1: {}            32: {1,1,1,1,1}       42: {1,2,4}
    2: {1}           13: {6}               44: {1,1,5}
    3: {2}           25: {3,3}             54: {1,2,2,2}
    4: {1,1}         21: {2,4}             60: {1,1,2,3}
    5: {3}           22: {1,5}             56: {1,1,1,4}
    6: {1,2}         27: {2,2,2}           72: {1,1,1,2,2}
    8: {1,1,1}       30: {1,2,3}           80: {1,1,1,1,3}
    7: {4}           28: {1,1,4}           96: {1,1,1,1,1,2}
    9: {2,2}         36: {1,1,2,2}        128: {1,1,1,1,1,1,1}
   10: {1,3}         40: {1,1,1,3}         19: {8}
   12: {1,1,2}       48: {1,1,1,1,2}       49: {4,4}
   16: {1,1,1,1}     64: {1,1,1,1,1,1}     55: {3,5}
   11: {5}           17: {7}               39: {2,6}
   15: {2,3}         35: {3,4}             34: {1,7}
   14: {1,4}         33: {2,5}             75: {2,3,3}
   18: {1,2,2}       26: {1,6}             63: {2,2,4}
   20: {1,1,3}       45: {2,2,3}           70: {1,3,4}
   24: {1,1,1,2}     50: {1,3,3}           66: {1,2,5}
Triangle begins:
   1
   2
   3   4
   5   6   8
   7   9  10  12  16
  11  15  14  18  20  24  32
  13  25  21  22  27  30  28  36  40  48  64
  17  35  33  26  45  50  42  44  54  60  56  72  80  96 128
This corresponds to the tetrangle:
                  0
                 (1)
               (2)(11)
             (3)(21)(111)
        (4)(22)(31)(211)(1111)
  (5)(32)(41)(221)(311)(2111)(11111)

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

Row lengths are A000041.
Compositions under the same order are A124734 (triangle).
The version for reversed (weakly increasing) partitions is A185974.
The constructive version is A334301.
Ignoring length gives A334434, or A334437 for reversed partitions.
The dual version (sum/length/revlex) is A334438.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colexicographic order (sum/length/colex) are A036037.
Graded reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Graded lexicographically ordered partitions are A193073.
Graded Heinz numbers are A215366.
Sorting partitions by Heinz number gives A296150.
Partitions in increasing-length reverse-lexicographic order (sum/length/revlex) are A334439 (not A036037).
Cf. A026791, A036043, A048793, A056239, A129129, A211992, A228351, A228531, A334302, A334435, A334436.

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

A001222(a(n)) = A036043(n).

A334302 Irregular triangle read by rows where row k is the k-th reversed integer partition, if reversed partitions are sorted first by sum, then by length, and finally reverse-lexicographically.

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

Offset: 0

Gus Wiseman, Apr 30 2020

			The sequence of all reversed partitions begins:
  ()         (1,4)        (1,1,1,1,2)
  (1)        (1,2,2)      (1,1,1,1,1,1)
  (2)        (1,1,3)      (7)
  (1,1)      (1,1,1,2)    (3,4)
  (3)        (1,1,1,1,1)  (2,5)
  (1,2)      (6)          (1,6)
  (1,1,1)    (3,3)        (2,2,3)
  (4)        (2,4)        (1,3,3)
  (2,2)      (1,5)        (1,2,4)
  (1,3)      (2,2,2)      (1,1,5)
  (1,1,2)    (1,2,3)      (1,2,2,2)
  (1,1,1,1)  (1,1,4)      (1,1,2,3)
  (5)        (1,1,2,2)    (1,1,1,4)
  (2,3)      (1,1,1,3)    (1,1,1,2,2)
This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows.
                            0
                           (1)
                        (2) (1,1)
                    (3) (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)
Showing partitions as their Heinz numbers (see A334435) gives:
   1
   2
   3   4
   5   6   8
   7   9  10  12  16
  11  15  14  18  20  24  32
  13  25  21  22  27  30  28  36  40  48  64
  17  35  33  26  45  50  42  44  54  60  56  72  80  96 128

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

Row lengths are A036043.
Lexicographically ordered reversed partitions are A026791.
The dual ordering (sum/length/lex) of reversed partitions is A036036.
Reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Lexicographically ordered partitions are A193073.
Graded Heinz numbers are A215366.
Ignoring length gives A228531.
Sorting partitions by Heinz number gives A296150.
The version for compositions is A296774.
The dual ordering (sum/length/lex) of non-reversed partitions is A334301.
Taking Heinz numbers gives A334435.
The version for regular (non-reversed) partitions is A334439 (not A036037).
Cf. A000041, A048793, A066099, A080576, A124734, A162247, A211992, A228100, A228351.

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

A374515 Irregular triangle read by rows where row n lists the leaders of anti-runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 31 2024

Anti-runs summing to n are counted by A003242(n).
The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.
The k-th composition in standard order (graded reverse-lexicographic, A066099) 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 gives a bijective correspondence between nonnegative integers and integer compositions.

			The maximal anti-runs of the 1234567th composition in standard order are ((3,2,1,2),(2,1,2,5,1),(1),(1)), so row 1234567 is (3,2,1,1).
The nonnegative integers, corresponding compositions, and leaders of anti-runs begin:
    0:      () -> ()        15: (1,1,1,1) -> (1,1,1,1)
    1:     (1) -> (1)       16:       (5) -> (5)
    2:     (2) -> (2)       17:     (4,1) -> (4)
    3:   (1,1) -> (1,1)     18:     (3,2) -> (3)
    4:     (3) -> (3)       19:   (3,1,1) -> (3,1)
    5:   (2,1) -> (2)       20:     (2,3) -> (2)
    6:   (1,2) -> (1)       21:   (2,2,1) -> (2,2)
    7: (1,1,1) -> (1,1,1)   22:   (2,1,2) -> (2)
    8:     (4) -> (4)       23: (2,1,1,1) -> (2,1,1)
    9:   (3,1) -> (3)       24:     (1,4) -> (1)
   10:   (2,2) -> (2,2)     25:   (1,3,1) -> (1)
   11: (2,1,1) -> (2,1)     26:   (1,2,2) -> (1,2)
   12:   (1,3) -> (1)       27: (1,2,1,1) -> (1,1)
   13: (1,2,1) -> (1)       28:   (1,1,3) -> (1,1)
   14: (1,1,2) -> (1,1)     29: (1,1,2,1) -> (1,1)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Row-leaders of nonempty rows are A065120.
Row-lengths are A333381.
Row-sums are A374516.
Positions of identical rows are A374519 (counted by A374517).
Positions of distinct (strict) rows are A374638 (counted by A374518).
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
- Run-compression is A373948 or A374251, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Six types of maximal runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A029931, A059893, A228351, A233564, A272919, A333213, A373949.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[First/@Split[stc[n],UnsameQ],{n,0,100}]

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

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 30 2013

The representation of the partitions (for fixed n) is as (weakly) increasing lists of parts, the order between individual partitions (for the same n) is (list-)reversed lexicographic; see examples. [Joerg Arndt, Sep 03 2013]
Also compositions in the triangle of A066099 that are in nondecreasing order.
The equivalent sequence for compositions (ordered partitions) is A066099.
Row n has length A006128(n).
Row sums give A066186.

			Illustration of initial terms:
---------------------------------
.                    Ordered
n  j     Diagram     partition
---------------------------------
.              _
1  1          |_|    1;
.            _ _
2  1        |  _|    2,
2  2        |_|_|    1, 1;
.          _ _ _
3  1      |  _ _|    3,
3  2      | |  _|    1, 2,
3  3      |_|_|_|    1, 1, 1;
.        _ _ _ _
4  1    |    _ _|    4,
4  2    |  _|_ _|    2, 2,
4  3    | |  _ _|    1, 3,
4  4    | | |  _|    1, 1, 2,
4  5    |_|_|_|_|    1, 1, 1, 1;
.
Triangle begins:
[1];
[2],[1,1];
[3],[1,2],[1,1,1];
[4],[2,2],[1,3],[1,1,2],[1,1,1,1];
[5],[2,3],[1,4],[1,2,2],[1,1,3],[1,1,1,2],[1,1,1,1,1];
[6],[3,3],[2,4],[2,2,2],[1,5],[1,2,3],[1,1,4],[1,1,2,2],[1,1,1,3],[1,1,1,1,2],[1,1,1,1,1,1];
[7],[3,4],[2,5],[2,2,3],[1,6],[1,3,3],[1,2,4],[1,2,2,2],[1,1,5],[1,1,2,3],[1,1,1,4],[1,1,1,2,2],[1,1,1,1,3],[1,1,1,1,1,2],[1,1,1,1,1,1,1];
...

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

Row lengths are A000041.
Partition sums are A036042.
Partition minima are A182715.
Partition lengths are A333486.
The lexicographic version (sum/lex) is A026791.
Compositions under the same order (sum/revlex) are A066099.
The colexicographic version (sum/colex) is A080576.
The version for non-reversed partitions is A080577.
The length-sensitive version (sum/length/revlex) is A334302.
The Heinz numbers of these partitions are A334436.
Partitions in colexicographic order (sum/colex) are A211992.
Partitions in lexicographic order (sum/lex) are A193073.
Cf. A026792, A036036, A049085, A103921, A112798, A115623, A129129, A228351, A331581, A334435, A334439, A334442.

revlexsort[f_,c_]:=OrderedQ[PadRight[{c,f}]];
Join@@Table[Sort[Reverse/@IntegerPartitions[n],revlexsort],{n,0,8}] (* Gus Wiseman, May 23 2020 *)

A080576 Triangle in which n-th row lists all partitions of n, in graded reflected lexicographic order.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 23 2003

The graded reflected lexicographic ordering of the partitions is used by Maple. - Daniel Forgues, Jan 19 2011
Each partition here is the conjugate of the corresponding partition in Abramowitz and Stegun order (A036036). The partitions are in the reverse of the order of the partitions in Mathematica order (A080577). - Franklin T. Adams-Watters, Oct 18 2006
Reversing all partitions gives A193073 (the non-reflected version). The version for reversed (weakly increasing) partitions is A211992. Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036. - Gus Wiseman, May 20 2020
Also reversed integer partitions in colexicographic order, cf. A228531. - Gus Wiseman, May 31 2020

			First five rows are:
[[1]]
[[1, 1], [2]]
[[1, 1, 1], [1, 2], [3]]
[[1, 1, 1, 1], [1, 1, 2], [2, 2], [1, 3], [4]]
[[1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 2, 2], [1, 1, 3], [2, 3], [1, 4], [5]]
From _Gus Wiseman_, May 20 2020: (Start)
The sequence of all reversed partitions begins:
  ()       (122)     (15)       (25)
  (1)      (113)     (6)        (16)
  (11)     (23)      (1111111)  (7)
  (2)      (14)      (111112)   (11111111)
  (111)    (5)       (11122)    (1111112)
  (12)     (111111)  (1222)     (111122)
  (3)      (11112)   (11113)    (11222)
  (1111)   (1122)    (1123)     (2222)
  (112)    (222)     (223)      (111113)
  (22)     (1113)    (133)      (11123)
  (13)     (123)     (1114)     (1223)
  (4)      (33)      (124)      (1133)
  (11111)  (114)     (34)       (233)
  (1112)   (24)      (115)      (11114)
(End)

Alois P. Heinz, Rows n = 1..20, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. (uses Flash)
A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions
Wikiversity, Lexicographic and colexicographic order

See A080577 for the Mathematica (graded reverse lexicographic) ordering.
See A036036 for the Hindenburg (graded reflected colexicographic) ordering (listed in the Abramowitz and Stegun Handbook).
See A036037 for the graded colexicographic ordering.
See A193073 for the graded lexicographic ordering. - M. F. Hasler, Jul 16 2011
See A228100 for the Fenner-Loizou (binary tree) ordering.
Row n has A000041(n) partitions.
Taking colexicographic instead of lexicographic gives A026791.
Lengths of these partitions appear to be A049085.
Reversing all partitions gives A193073 (the non-reflected version).
The version for reversed (weakly increasing) partitions is A211992.
The generalization to compositions is A228525.
The Heinz numbers of these partitions are A334434.
Cf. A026791, A036037, A112798, A129129, A185974, A228351, A228531, A334301, A334302, A334433, A334437.

Maple
```
with(combinat); partition(6);
```

row[n_] := Flatten[Reverse /@ Reverse[SplitBy[Reverse /@ IntegerPartitions[n], Length]], 1]; Array[row, 7] // Flatten (* Jean-François Alcover, Dec 05 2016 *)
lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Reverse/@Join@@Table[Sort[IntegerPartitions[n],lexsort],{n,0,8}] (* Gus Wiseman, May 20 2020 *)

Edited by Daniel Forgues, Jan 21 2011

A124758 Product of the parts of the compositions in standard order.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 2, 1, 4, 3, 4, 2, 3, 2, 2, 1, 5, 4, 6, 3, 6, 4, 4, 2, 4, 3, 4, 2, 3, 2, 2, 1, 6, 5, 8, 4, 9, 6, 6, 3, 8, 6, 8, 4, 6, 4, 4, 2, 5, 4, 6, 3, 6, 4, 4, 2, 4, 3, 4, 2, 3, 2, 2, 1, 7, 6, 10, 5, 12, 8, 8, 4, 12, 9, 12, 6, 9, 6, 6, 3, 10, 8, 12, 6, 12, 8, 8, 4, 8, 6, 8, 4, 6, 4, 4, 2, 6, 5, 8, 4, 9, 6

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) 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. - Gus Wiseman, Apr 03 2020

			Composition number 11 is 2,1,1; 2*1*1 = 2, so a(11) = 2.
The table starts:
  1
  1
  2 1
  3 2 2 1
  4 3 4 2 3 2 2 1
  5 4 6 3 6 4 4 2 4 3 4 2 3 2 2 1
The 146-th composition in standard order is (3,3,2), with product 18, so a(146) = 18. - _Gus Wiseman_, Apr 03 2020

Alois P. Heinz, Rows n = 0..14, flattened
George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021.

Cf. A066099, A118851, A011782 (row lengths), A001906 (row sums).
The lengths of standard compositions are given by A000120.
The version for prime indices is A003963.
The version for binary indices is A096111.
Taking the sum instead of product gives A070939.
The sum of binary indices is A029931.
The sum of prime indices is A056239.
Taking GCD instead of product gives A326674.
Positions of first appearances are A331579.
Cf. A001519, A011782, A048793, A114994, A124767, A228351, A272919, A329369 (similar recurrence), A333218, A333219.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Times@@stc[n],{n,0,100}] (* Gus Wiseman, Apr 03 2020 *)

For a composition b(1),...,b(k), a(n) = Product_{i=1}^k b(i).
a(A164894(n)) = a(A246534(n)) = n!. - Gus Wiseman, Apr 03 2020
a(A233249(n)) = a(A333220(n)) = A003963(n). - Gus Wiseman, Apr 03 2020
From Mikhail Kurkov, Jul 11 2021: (Start)
Some conjectures:
a(2n+1) = a(n) for n >= 0.
a(2n) = (1 + 1/A001511(n))*a(n) = 2*a(n) + a(n - 2^f(n)) - a(2n - 2^f(n)) for n > 0 with a(0)=1 where f(n) = A007814(n).
From the 1st formula for a(2n) we get a(4n+2) = 2*a(n), a(4n) = 2*a(2n) - a(n).
Sum_{k=0..2^n - 1} a(k) = A001519(n+1) for n >= 0.
a((4^n - 1)/3) = A011782(n) for n >= 0.
a(2^m*(2^n - 1)) = m + 1 for n > 0, m >= 0. (End)

A228525 Triangle read by rows in which row n lists the compositions (ordered partitions) of n in colexicographic order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 24 2013

The representation of the compositions (for fixed n) is as lists of parts, the order between individual compositions (for the same n) is co-lexicographic. [Joerg Arndt, Sep 02 2013]
The equivalent sequence for partitions is A211992.
Row n has length A001792(n-1).
Row sums give A001787, n >= 1.

			Illustration of initial terms:
---------------------------------
n  j     Diagram     Composition
---------------------------------
.         _
1  1     |_|         1;
.         _ _
2  1     |_| |       1, 1,
2  2     |_ _|       2;
.         _ _ _
3  1     |_| | |     1, 1, 1,
3  2     |_ _| |     2, 1,
3  3     |_|   |     1, 2,
3  4     |_ _ _|     3;
.         _ _ _ _
4  1     |_| | | |   1, 1, 1, 1,
4  2     |_ _| | |   2, 1, 1,
4  3     |_|   | |   1, 2, 1,
4  4     |_ _ _| |   3, 1,
4  5     |_| |   |   1, 1, 2,
4  6     |_ _|   |   2, 2,
4  7     |_|     |   1, 3,
4  8     |_ _ _ _|   4;
.
Triangle begins:
[1];
[1,1],[2];
[1,1,1],[2,1],[1,2],[3];
[1,1,1,1],[2,1,1],[1,2,1],[3,1],[1,1,2],[2,2],[1,3],[4];
[1,1,1,1,1],[2,1,1,1],[1,2,1,1],[3,1,1],[1,1,2,1],[2,2,1],[1,3,1],[4,1],[1,1,1,2],[2,1,2],[1,2,2],[3,2],[1,1,3],[2,3],[1,4],[5];

Joerg Arndt, Table of n, a(n) for n = 1..10000

Cf. A066099, A211992, A228351, A228369.

gen_comp(n)=
{  /* Generate compositions of n as lists of parts (order is lex): */
    my(ct = 0);
    my(m, z, pt);
    \\ init:
    my( a = vector(n, j, 1) );
    m = n;
    while ( 1,
        ct += 1;
        pt = vector(m, j, a[j]);
\\        /* for A228369  print composition: */
\\        for (j=1, m, print1(pt[j],", ") );
        /* for A228525 print reversed (order is colex): */
        forstep (j=m, 1, -1, print1(pt[j],", ") );
        if ( m<=1,  return(ct) );  \\ current is last
        a[m-1] += 1;
        z = a[m] - 2;
        a[m] = 1;
        m += z;
    );
    return(ct);
}
for(n=1, 12, gen_comp(n) );
\\ Joerg Arndt, Sep 02 2013

A348612 Numbers k such that the k-th composition in standard order is not an anti-run, i.e., has adjacent equal parts.

Original entry on oeis.org

3, 7, 10, 11, 14, 15, 19, 21, 23, 26, 27, 28, 29, 30, 31, 35, 36, 39, 42, 43, 46, 47, 51, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 73, 74, 75, 78, 79, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 99, 100, 103, 106, 107, 110, 111, 112, 113, 114, 115, 116

Offset: 1

Gus Wiseman, Nov 03 2021

nonn

First differs from A345168 in lacking 37, corresponding to the composition (3,2,1).

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) 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.

			The terms and corresponding standard compositions begin:
     3: (1,1)          35: (4,1,1)        61: (1,1,1,2,1)
     7: (1,1,1)        36: (3,3)          62: (1,1,1,1,2)
    10: (2,2)          39: (3,1,1,1)      63: (1,1,1,1,1,1)
    11: (2,1,1)        42: (2,2,2)        67: (5,1,1)
    14: (1,1,2)        43: (2,2,1,1)      71: (4,1,1,1)
    15: (1,1,1,1)      46: (2,1,1,2)      73: (3,3,1)
    19: (3,1,1)        47: (2,1,1,1,1)    74: (3,2,2)
    21: (2,2,1)        51: (1,3,1,1)      75: (3,2,1,1)
    23: (2,1,1,1)      53: (1,2,2,1)      78: (3,1,1,2)
    26: (1,2,2)        55: (1,2,1,1,1)    79: (3,1,1,1,1)
    27: (1,2,1,1)      56: (1,1,4)        83: (2,3,1,1)
    28: (1,1,3)        57: (1,1,3,1)      84: (2,2,3)
    29: (1,1,2,1)      58: (1,1,2,2)      85: (2,2,2,1)
    30: (1,1,1,2)      59: (1,1,2,1,1)    86: (2,2,1,2)
    31: (1,1,1,1,1)    60: (1,1,1,3)      87: (2,2,1,1,1)

Constant run compositions are counted by A000005, ranked by A272919.
Counting these compositions by sum and length gives A131044.
These compositions are counted by A261983.
The complement is A333489, counted by A003242.
The non-alternating case is A345168, complement A345167.
A011782 counts compositions, strict A032020.
A238279 counts compositions by sum and number of maximal runs.
A274174 counts compositions with equal parts contiguous.
A336107 counts non-anti-run permutations of prime factors.
A345195 counts non-alternating anti-runs, ranked by A345169.
For compositions in standard order (rows of A066099):
- Length is A000120.
- Sum is A070939
- Maximal runs are counted by A124767.
- Strict compositions are ranked by A233564.
- Maximal anti-runs are counted by A333381.
- Runs-resistance is A333628.
Cf. A029931, A048793, A106356, A114901, A167606, A178470, A228351, A244164, A262046, A335452, A335464.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[100],MatchQ[stc[#],{_,x_,x_,_}]&]

A228100 Triangle in which n-th row lists all partitions of n, such that partitions of n into m parts appear in lexicographic order previous to the partitions of n into k parts if k < m. (Fenner-Loizou tree.)

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

Offset: 1

Peter Luschny, Aug 10 2013

First differs from A193073 at a(58). - Omar E. Pol, Sep 22 2013

The partition lengths appear to be A331581. - Gus Wiseman, May 12 2020

			The sixth row is:
[1, 1, 1, 1, 1, 1]
[2, 1, 1, 1, 1]
[2, 2, 1, 1]
[3, 1, 1, 1]
[2, 2, 2]
[3, 2, 1]
[4, 1, 1]
[3, 3]
[4, 2]
[5, 1]
[6]
From _Gus Wiseman_, May 10 2020: (Start)
The triangle with partitions shown as Heinz numbers (A333485) 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
(End)

T. I. Fenner, G. Loizou: A binary tree representation and related algorithms for generating integer partitions. The Computer J. 23(4), 332-337 (1980)
D. E. Knuth: The Art of Computer Programming. Generating all combinations and partitions, vol. 4, fasc. 3, 7.2.1.4, exercise 10.
K. Yamanaka, Y. Otachi, Sh. Nakano: Efficient enumeration of ordered trees with k leaves. In: WALCOM: Algorithms and Computation, Lecture Notes in Computer Science Volume 5431, 141-150 (2009)
S. Zaks, D. Richards: Generating trees and other combinatorial objects lexicographically. SIAM J. Comput. 8(1), 73-81 (1979)
A. Zoghbi, I. Stojmenovic': Fast algorithms for generating integer partitions. Int. J. Comput. Math. 70, 319-332 (1998)

Alois P. Heinz, rows n = 1..19, flattened
Peter Luschny, Integer Partition Trees
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

See A036036 for the Hindenburg (graded reflected colexicographic) ordering.
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 A182937 the Fenner-Loizou (binary tree in preorder traversal) ordering.
See A193073 for the graded lexicographic ordering.
The version for compositions is A296773.
Taking Heinz numbers gives A333485.
Lexicographically ordered reversed partitions are A026791.
Sorting partitions by Heinz number gives A296150, or A112798 for reversed partitions.
Reversed partitions under the (sum/length/revlex) ordering are A334302.
Cf. A000041, A036043, A048793, A211992, A228351, A228531, A331581, A333484, A334301, A334439.

b:= proc(n, i) b(n, i):= `if`(n=0 or i=1, [[1$n]], [b(n, i-1)[],
      `if`(i>n, [], map(x-> [i, x[]], b(n-i, i)))[]])
    end:
T:= n-> map(h-> h[], sort(b(n$2), proc(x, y) local i;
        if nops(x)<>nops(y) then return nops(x)>nops(y) else
        for i to nops(x) do if x[i]<>y[i] then return x[i]Alois P. Heinz, Aug 13 2013

row[n_] := Flatten[Reverse[Sort[#]]& /@ SplitBy[Sort[IntegerPartitions[n] ], Length], 1] // Reverse; Array[row, 8] // Flatten (* Jean-François Alcover, Dec 05 2016 *)
ralensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]>Length[c],OrderedQ[{f,c}]];
Join@@Table[Sort[IntegerPartitions[n],ralensort],{n,0,8}] (* Gus Wiseman, May 10 2020 *)

from collections import deque
def GeneratePartitions(n, visit):
    p = ([], 0, n)
    queue = deque()
    queue.append(p)
    visit(p)
    while len(queue) > 0 :
        (phead, pheadLen, pnum1s) = queue.popleft()
        if pnum1s != 1 :
            head = phead[:pheadLen] + [2]
            q = (head, pheadLen + 1, pnum1s - 2)
            if 1 <= q[2] : queue.append(q)
            visit(q)
        if pheadLen == 1 or (pheadLen > 1 and \
                      (phead[pheadLen - 1] != phead[pheadLen - 2])) :
            head = phead[:pheadLen]
            head[pheadLen - 1] += 1
            q = (head, pheadLen, pnum1s - 1)
            if 1 <= q[2] : queue.append(q)
            visit(q)
def visit(q): print(q[0] + [1 for i in range(q[2])])
for n in (1..7): GeneratePartitions(n, visit)

A333628 Runs-resistance of the n-th composition in standard order. Number of steps taking run-lengths to reduce the n-th composition in standard order to a singleton.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 1, 0, 2, 1, 3, 2, 2, 3, 1, 0, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 4, 3, 4, 3, 1, 0, 2, 2, 3, 1, 2, 2, 3, 2, 2, 1, 2, 2, 2, 3, 3, 2, 2, 2, 4, 2, 3, 2, 4, 3, 4, 2, 3, 3, 4, 3, 1, 0, 2, 2, 3, 2, 2, 2, 3, 2, 3, 3, 4, 2, 2, 3, 3, 2, 2, 2, 4, 3, 3, 4

Offset: 1

Gus Wiseman, Mar 31 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) 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.

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			Starting with 13789 and repeatedly applying A333627 gives: 13789 -> 859 -> 110 -> 29 -> 11 -> 6 -> 3 -> 2, corresponding to the compositions: (1,2,2,1,1,2,1,1,2,1) -> (1,2,2,1,2,1,1) -> (1,2,1,1,2) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1) -> (2), so a(13789) = 7.

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Number of times applying A333627 to reach a power of 2, starting with n.
Positions of first appearances are A333629.
All of the following pertain to compositions in standard order (A066099):
- The length is A000120.
- The partial sums from the right are A048793.
- The sum is A070939.
- Adjacent equal pairs are counted by A124762.
- Equal runs are counted by A124767.
- Strict compositions are ranked by A233564.
- The partial sums from the left are A272020.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Heinz number is A333219.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
- First appearances for specified run-lengths are A333630.
Cf. A029931, A098504, A114994, A225620, A228351, A238279, A242882, A318928, A329744, A329747, A333489.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[runsres[stc[n]],{n,100}]

cp's OEIS Frontend

A334433 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally lexicographically.

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A334302 Irregular triangle read by rows where row k is the k-th reversed integer partition, if reversed partitions are sorted first by sum, then by length, and finally reverse-lexicographically.

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A374515 Irregular triangle read by rows where row n lists the leaders of anti-runs in the n-th composition in standard order.

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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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A080576 Triangle in which n-th row lists all partitions of n, in graded reflected lexicographic order.

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Maple

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A124758 Product of the parts of the compositions in standard order.

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A228525 Triangle read by rows in which row n lists the compositions (ordered partitions) of n in colexicographic order.

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PARI

A348612 Numbers k such that the k-th composition in standard order is not an anti-run, i.e., has adjacent equal parts.

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