A228351 -id:A228351 - cp's OEIS frontend

A353432 Numbers k such that the k-th composition in standard order has its own run-lengths as a consecutive subsequence.

0, 1, 10, 21, 26, 43, 58, 107, 117, 174, 186, 292, 314, 346, 348, 349, 373, 430, 442, 570, 585, 586, 629, 676, 696, 697, 804, 826, 860, 861, 885, 1082, 1141, 1173, 1210, 1338, 1387, 1392, 1393, 1394, 1396, 1594, 1653, 1700, 1720, 1721, 1882, 2106, 2165, 2186

Offset: 1

Gus Wiseman, May 16 2022

nonn

First differs from A353402 (the non-consecutive version) in lacking 53.

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 initial terms, their binary expansions, and the corresponding standard compositions:
     0:          0  ()
     1:          1  (1)
    10:       1010  (2,2)
    21:      10101  (2,2,1)
    26:      11010  (1,2,2)
    43:     101011  (2,2,1,1)
    58:     111010  (1,1,2,2)
   107:    1101011  (1,2,2,1,1)
   117:    1110101  (1,1,2,2,1)
   174:   10101110  (2,2,1,1,2)
   186:   10111010  (2,1,1,2,2)
   292:  100100100  (3,3,3)
   314:  100111010  (3,1,1,2,2)
   346:  101011010  (2,2,1,2,2)
   348:  101011100  (2,2,1,1,3)
   349:  101011101  (2,2,1,1,2,1)
   373:  101110101  (2,1,1,2,2,1)
   430:  110101110  (1,2,2,1,1,2)
   442:  110111010  (1,2,1,1,2,2)

These compositions are counted by A353392.
This is the consecutive case of A353402, counted by A353390.
The non-consecutive recursive version is A353431, counted by A353391.
The recursive version is A353696, counted by A353430.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, rev A228351, run-lens A333769.
A329738 counts uniform compositions, partitions A047966.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767, distinct A351014.
- Subsequences are counted by A334299, contiguous A124770/A124771.
- Runs-resistance is A333628.
Classes of standard compositions:
- Partitions are A114994, strict A333255, rev A225620, strict rev A333256.
- Runs are A272919, counted by A000005.
- Golomb rulers are A333222, counted by A169942.
- Anti-runs are A333489, counted by A003242.
Cf. A044813, A165413, A181819, A318928, A325702, A325705, A325755, A333224, A333755, A353389, A353393, A353403.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
rorQ[y_]:=Length[y]==0||MemberQ[Join@@Table[Take[y,{i,j}],{i,Length[y]},{j,i,Length[y]}],Length/@Split[y]];
Select[Range[0,10000],rorQ[stc[#]]&]

A374520 Numbers k such that the leaders of maximal anti-runs in the k-th composition in standard order (A066099) are not identical.

Original entry on oeis.org

11, 19, 23, 26, 35, 39, 43, 46, 47, 53, 58, 67, 71, 74, 75, 78, 79, 83, 87, 91, 92, 93, 94, 95, 100, 106, 107, 117, 122, 131, 135, 138, 139, 142, 143, 147, 149, 151, 154, 155, 156, 157, 158, 159, 163, 164, 167, 171, 174, 175, 179, 183, 184, 185, 186, 187, 188

Offset: 1

Gus Wiseman, Aug 06 2024

nonn

The leaders of maximal 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 sequence together with corresponding compositions begins:
  11: (2,1,1)
  19: (3,1,1)
  23: (2,1,1,1)
  26: (1,2,2)
  35: (4,1,1)
  39: (3,1,1,1)
  43: (2,2,1,1)
  46: (2,1,1,2)
  47: (2,1,1,1,1)
  53: (1,2,2,1)
  58: (1,1,2,2)
  67: (5,1,1)
  71: (4,1,1,1)
  74: (3,2,2)
  75: (3,2,1,1)
  78: (3,1,1,2)
  79: (3,1,1,1,1)
  83: (2,3,1,1)
  87: (2,2,1,1,1)
  91: (2,1,2,1,1)

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

For leaders of maximal constant runs we have the complement of A272919.
Positions of non-constant rows in A374515.
The complement is A374519, counted by A374517.
For distinct instead of identical leaders we have A374639, counted by A374678, complement A374638, counted by A374518.
Compositions of this type are counted by A374640.
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
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 transform is A373948, 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, A228351, A233564, A238343, A238424, A333213, A373949.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!SameQ@@First/@Split[stc[#],UnsameQ]&]

A124763 Number of non-rises (levels or falls) for compositions in standard order.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 2, 0, 1, 1, 2, 0, 1, 1, 3, 0, 1, 1, 2, 0, 2, 1, 3, 0, 1, 1, 2, 1, 2, 2, 4, 0, 1, 1, 2, 1, 2, 1, 3, 0, 1, 2, 3, 1, 2, 2, 4, 0, 1, 1, 2, 0, 2, 1, 3, 1, 2, 2, 3, 2, 3, 3, 5, 0, 1, 1, 2, 1, 2, 1, 3, 0, 2, 2, 3, 1, 2, 2, 4, 0, 1, 1, 2, 1, 3, 2, 4, 1, 2, 2, 3, 2, 3, 3, 5, 0, 1, 1, 2, 1, 2, 1, 3, 0

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. a(n) is one fewer than the number of maximal strictly increasing runs in this composition. Alternatively, a(n) is the number of weak descents in the same composition. For example, the strictly increasing runs of the 1234567th composition are ((3),(2),(1,2),(2),(1,2,5),(1),(1),(1)), so a(1234567) = 8 - 1 = 7. The 7 weak descents together with the strict ascents are: 3 >= 2 >= 1 < 2 >= 2 >= 1 < 2 < 5 >= 1 >= 1 >= 1. - Gus Wiseman, Apr 08 2020

			Composition number 11 is 2,1,1; 2>=1>=1, so a(11) = 2.
The table starts:
  0
  0
  0 1
  0 1 0 2
  0 1 1 2 0 1 1 3
  0 1 1 2 0 2 1 3 0 1 1 2 1 2 2 4
  0 1 1 2 1 2 1 3 0 1 2 3 1 2 2 4 0 1 1 2 0 2 1 3 1 2 2 3 2 3 3 5

Cf. A029931, A066099, A124760, A124761, A124764, A011782 (row lengths), A045883 (row sums), A238343, A333220.
Compositions of n with k weak descents are A333213.
Positions of zeros are A333255.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Partial sums from the right are A048793.
- Sum is A070939.
- Weakly decreasing compositions are A114994.
- Adjacent equal pairs are counted by A124762.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Weakly increasing compositions are A225620.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Initial intervals are A246534.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Permutations are A333218.
- Heinz number is A333219.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Anti-runs are A333489.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Select[Partition[stc[n],2,1],GreaterEqual@@#&]],{n,0,100}] (* Gus Wiseman, Apr 08 2020 *)

For a composition b(1),...,b(k), a(n) = Sum_{1<=i=1=b(i+1)} 1.
a(n) = A124761(n) + A124762(n).
For n > 0, a(n) = A124768(n) - 1. - Gus Wiseman, Apr 08 2020

A333379 Numbers k such that the k-th composition in standard order is weakly increasing and covers an initial interval of positive integers.

Original entry on oeis.org

0, 1, 3, 6, 7, 14, 15, 26, 30, 31, 52, 58, 62, 63, 106, 116, 122, 126, 127, 212, 234, 244, 250, 254, 255, 420, 426, 468, 490, 500, 506, 510, 511, 840, 852, 932, 938, 980, 1002, 1012, 1018, 1022, 1023, 1700, 1706, 1864, 1876, 1956, 1962, 2004, 2026, 2036, 2042

Offset: 1

Gus Wiseman, Mar 21 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.

			The sequence of terms together with the corresponding compositions begins:
    0: ()               127: (1,1,1,1,1,1,1)
    1: (1)              212: (1,2,2,3)
    3: (1,1)            234: (1,1,2,2,2)
    6: (1,2)            244: (1,1,1,2,3)
    7: (1,1,1)          250: (1,1,1,1,2,2)
   14: (1,1,2)          254: (1,1,1,1,1,1,2)
   15: (1,1,1,1)        255: (1,1,1,1,1,1,1,1)
   26: (1,2,2)          420: (1,2,3,3)
   30: (1,1,1,2)        426: (1,2,2,2,2)
   31: (1,1,1,1,1)      468: (1,1,2,2,3)
   52: (1,2,3)          490: (1,1,1,2,2,2)
   58: (1,1,2,2)        500: (1,1,1,1,2,3)
   62: (1,1,1,1,2)      506: (1,1,1,1,1,2,2)
   63: (1,1,1,1,1,1)    510: (1,1,1,1,1,1,1,2)
  106: (1,2,2,2)        511: (1,1,1,1,1,1,1,1,1)
  116: (1,1,2,3)        840: (1,2,3,4)
  122: (1,1,1,2,2)      852: (1,2,2,2,3)
  126: (1,1,1,1,1,2)    932: (1,1,2,3,3)

Sequences covering an initial interval are counted by A000670.
Compositions in standard order are A066099.
Weakly increasing runs are counted by A124766.
Removing the covering condition gives A225620.
Removing the ordering condition gives A333217.
The strictly increasing case is A164894.
The strictly decreasing version is A246534.
The unequal version is A333218.
The weakly decreasing version is A333380.
Cf. A000120, A000225, A029931, A048793, A070939, A228351, A233564, A272919.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],normQ[stc[#]]&&LessEqual@@stc[#]&]

Intersection of A333217 and A225620.

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

A334266 Numbers k such that the k-th composition in standard order is both a reversed Lyndon word and a co-Lyndon word.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 11, 16, 17, 18, 19, 21, 23, 32, 33, 34, 35, 37, 39, 43, 47, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 85, 87, 91, 95, 128, 129, 130, 131, 132, 133, 135, 137, 138, 139, 141, 143, 146, 147, 149, 151, 155, 159, 171, 173, 175, 183, 191

Offset: 1

Gus Wiseman, Apr 22 2020

nonn

A Lyndon word is a finite sequence of positive integers that is lexicographically strictly less than all of its cyclic rotations. Co-Lyndon is defined similarly, except with strictly greater instead of strictly less.

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 sequence of all reversed Lyndon co-Lyndon words begins:
    0: ()            37: (3,2,1)         91: (2,1,2,1,1)
    1: (1)           39: (3,1,1,1)       95: (2,1,1,1,1,1)
    2: (2)           43: (2,2,1,1)      128: (8)
    4: (3)           47: (2,1,1,1,1)    129: (7,1)
    5: (2,1)         64: (7)            130: (6,2)
    8: (4)           65: (6,1)          131: (6,1,1)
    9: (3,1)         66: (5,2)          132: (5,3)
   11: (2,1,1)       67: (5,1,1)        133: (5,2,1)
   16: (5)           68: (4,3)          135: (5,1,1,1)
   17: (4,1)         69: (4,2,1)        137: (4,3,1)
   18: (3,2)         71: (4,1,1,1)      138: (4,2,2)
   19: (3,1,1)       73: (3,3,1)        139: (4,2,1,1)
   21: (2,2,1)       74: (3,2,2)        141: (4,1,2,1)
   23: (2,1,1,1)     75: (3,2,1,1)      143: (4,1,1,1,1)
   32: (6)           77: (3,1,2,1)      146: (3,3,2)
   33: (5,1)         79: (3,1,1,1,1)    147: (3,3,1,1)
   34: (4,2)         85: (2,2,2,1)      149: (3,2,2,1)
   35: (4,1,1)       87: (2,2,1,1,1)    151: (3,2,1,1,1)

The version for binary expansion is A334267.
Compositions of this type are counted by A334269.
Normal sequences of this type are counted by A334270.
Necklace compositions of this type are counted by A334271.
Binary Lyndon words are counted by A001037.
Lyndon compositions are counted by A059966.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Reversed Lyndon words are A334265.
- Co-Lyndon words are A326774.
- Reversed co-Lyndon words are A328596.
- Length of Lyndon factorization is A329312.
- Length of Lyndon factorization of reverse is A334297.
- Length of co-Lyndon factorization is A334029.
- Length of co-Lyndon factorization of reverse is A329313.
- Distinct rotations are counted by A333632.
- Co-Lyndon factorizations are counted by A333765.
- Lyndon factorizations are counted by A333940.
Cf. A000740, A008965, A065609, A329324, A333764, A333943, A334272, A334297.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
Select[Range[0,100],lynQ[Reverse[stc[#]]]&&colynQ[stc[#]]&]

Intersection of A334265 and A326774.

A344089 Flattened tetrangle of reversed strict integer partitions, sorted first by length and then colexicographically.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 12 2021

First differs from the revlex (instead of colex) version for partitions of 12.
The zeroth row contains only the empty partition.
A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (3)(12)
  4: (4)(13)
  5: (5)(23)(14)
  6: (6)(24)(15)(123)
  7: (7)(34)(25)(16)(124)
  8: (8)(35)(26)(17)(134)(125)
  9: (9)(45)(36)(27)(18)(234)(135)(126)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724 plus one.
Taking lex instead of colex gives A026793 (non-reversed: A118457).
Triangle sums are A066189.
Reversing all partitions gives A344090.
The non-strict version is A344091.
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
Cf. A005117, A014466, A209862, A325683, A325859.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A246688, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344087, A344088, A344089.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Table[Reverse/@Sort[Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,30}]

A163510 Irregular table read by rows: Write n in binary. For each 1, the m-th term of row n is the number of 0's between the m-th 1, reading right to left, and the (m-1)th 1 (or the right side of the number if m-1 = 0).

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 1, 1, 0, 0, 0, 0, 3, 0, 2, 1, 1, 0, 0, 1, 2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 4, 0, 3, 1, 2, 0, 0, 2, 2, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 3, 0, 0, 2, 0, 1, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 4, 1, 3, 0, 0, 3, 2, 2, 0, 1, 2, 1, 0, 2, 0, 0, 0, 2, 3, 1, 0, 2, 1, 1, 1, 1, 0, 0, 1, 1, 2, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1

Offset: 1

Leroy Quet, Jul 29 2009

Row n contains exactly A000120(n) terms, for each n.
All odd-numbered rows begin with 0. All even-numbered rows begin with a positive integer.
Can be used to compute the permutation A163511.

			Table begins as:
  Row  n in    Terms on
   n   binary  that row
   1      1    0; (the distance of 1-bit from the right edge is zero)
   2     10    1; (the distance of 1-bit from the right edge is one)
   3     11    0,0;
   4    100    2;
   5    101    0,1; (the least significant 1-bit is zero steps away from the right edge, and there is one zero between those two 1-bits)
   6    110    1,0;
   7    111    0,0,0;
   8   1000    3;
   9   1001    0,2;
  10   1010    1,1;
  11   1011    0,0,1;
  12   1100    2,0;
  13   1101    0,1,0;
  14   1110    1,0,0;
  15   1111    0,0,0,0;
  16  10000    4;

Antti Karttunen, Table of n, a(n) for n = 1..11265

Cf. A000120, A006068, A100922, A227186, A243067, A163511, A227736.

Equals A228351-1, termwise.

Table[Reverse@ Map[Ceiling[(Length@ # - 1)/2] &, DeleteCases[Split@ Join[Riffle[IntegerDigits[n, 2], 0], {0}], {k__} /; k == 1]], {n, 46}] // Flatten (* Michael De Vlieger, Jul 25 2016 *)

from itertools import count, islice
def A163510_gen(): # generator of terms
    for n in count(1):
        k = n
        while k:
            yield (s:=(~k&k-1).bit_length())
            k >>= s+1
A163510_list = list(islice(A163510_gen(),30)) # Chai Wah Wu, Jul 17 2023

(define (A163510 n) (- (A227186bi (A006068 (A100922 (- n 1))) (A243067 n)) 1))
;; See A227186 for A227186bi. - Antti Karttunen, Jun 19 2014

a(n) = A227186(A006068(A100922(n-1)), A243067(n)) - 1. - Antti Karttunen, Jun 19 2014

Additional terms computed and Example section added by Antti Karttunen, Jun 19 2014

A228350 Triangle read by rows: T(j,k) is the k-th part in nonincreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 20 2013

Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A065120, n >= 1.

The equivalent sequence for integer partitions is A206437.

			---------------------------------------------------------
.              Diagram                Triangle
Compositions     of            of compositions (rows)
.   of 5       regions          and regions (columns)
----------------------------------------------------------
.             _ _ _ _ _
.         5  |_        |                                5
.       1+4  |_|_      |                              1 4
.       2+3  |_  |     |                            2   3
.     1+1+3  |_|_|_    |                          1 1   3
.       3+2  |_    |   |                        3       2
.     1+2+2  |_|_  |   |                      1 2       2
.     2+1+2  |_  | |   |                    2   1       2
.   1+1+1+2  |_|_|_|_  |                  1 1   1       2
.       4+1  |_      | |                4               1
.     1+3+1  |_|_    | |              1 3               1
.     2+2+1  |_  |   | |            2   2               1
.   1+1+2+1  |_|_|_  | |          1 1   2               1
.     3+1+1  |_    | | |        3       1               1
.   1+2+1+1  |_|_  | | |      1 2       1               1
.   2+1+1+1  |_  | | | |    2   1       1               1
. 1+1+1+1+1  |_|_|_|_|_|  1 1   1       1               1
.
Also the structure could be represented by an isosceles triangle in which the n-th diagonal gives the n-th region. For the composition of 4 see below:
.             _ _ _ _
.         4  |_      |                  4
.       1+3  |_|_    |                1   3
.       2+2  |_  |   |              2       2
.     1+1+2  |_|_|_  |            1   1       2
.       3+1  |_    | |          3               1
.     1+2+1  |_|_  | |        1   2               1
.     2+1+1  |_  | | |      2       1               1
.   1+1+1+1  |_|_|_|_|    1   1       1               1
.
Illustration of the four sections of the set of compositions of 4:
.                                      _ _ _ _
.                                     |_      |     4
.                                     |_|_    |   1+3
.                                     |_  |   |   2+2
.                       _ _ _         |_|_|_  | 1+1+2
.                      |_    |   3          | |     1
.             _ _      |_|_  | 1+2          | |     1
.     _      |_  | 2       | |   1          | |     1
.    |_| 1     |_| 1       |_|   1          |_|     1
.
.
Illustration of initial terms. The parts of the eight regions of the set of compositions of 4:
--------------------------------------------------------
\j:  1      2    3        4     5      6    7          8
k
--------------------------------------------------------
.  _    _ _    _    _ _ _     _    _ _    _    _ _ _ _
1 |_|1 |_  |2 |_|1 |_    |3  |_|1 |_  |2 |_|1 |_      |4
2        |_|1        |_  |2         |_|1        |_    |3
3                      | |1                       |   |2
4                      |_|1                       |_  |2
5                                                   | |1
6                                                   | |1
7                                                   | |1
8                                                   |_|1
.
Triangle begins:
1;
2,1;
1;
3,2,1,1;
1;
2,1;
1;
4,3,2,2,1,1,1,1;
1;
2,1;
1;
3,2,1,1;
1;
2,1;
1;
5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1;
...
.
Also triangle read by rows T(n,m) in which row n lists the parts of the n-th section of the set of compositions of the integers >= n, ordered by regions. Row lengths give A045623. Row sums give A001792 (see below):
[1];
[2,1];
[1],[3,2,1,1];
[1],[2,1],[1],[4,3,2,2,1,1,1,1];
[1],[2,1],[1],[3,2,1,1],[1],[2,1],[1],[5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1];

Main triangle: column 1 is A001511. Row j has length A006519(j). Row sums give A038712.

Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A135010, A141285, A187816, A187818, A193870, A206437, A228347, A228348, A228349, A228351, A228366, A228367, A228370, A228371, A228525, A228526.

T(j,k) = A065120(A001511(j)),k) = A001511(j) - A029837(k), 1<=k<=A006519(j), j>=1.

A335122 Irregular triangle whose reversed rows are all integer partitions in graded reverse-lexicographic order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 24 2020

First differs from A036036 for partitions of 6.
First differs from A334442 for partitions of 6.
Also reversed partitions in reverse-colexicographic order.

			The sequence of all reversed partitions begins:
  ()         (1,1,3)        (7)              (8)
  (1)        (1,2,2)        (1,6)            (1,7)
  (2)        (1,1,1,2)      (2,5)            (2,6)
  (1,1)      (1,1,1,1,1)    (1,1,5)          (1,1,6)
  (3)        (6)            (3,4)            (3,5)
  (1,2)      (1,5)          (1,2,4)          (1,2,5)
  (1,1,1)    (2,4)          (1,1,1,4)        (1,1,1,5)
  (4)        (1,1,4)        (1,3,3)          (4,4)
  (1,3)      (3,3)          (2,2,3)          (1,3,4)
  (2,2)      (1,2,3)        (1,1,2,3)        (2,2,4)
  (1,1,2)    (1,1,1,3)      (1,1,1,1,3)      (1,1,2,4)
  (1,1,1,1)  (2,2,2)        (1,2,2,2)        (1,1,1,1,4)
  (5)        (1,1,2,2)      (1,1,1,2,2)      (2,3,3)
  (1,4)      (1,1,1,1,2)    (1,1,1,1,1,2)    (1,1,3,3)
  (2,3)      (1,1,1,1,1,1)  (1,1,1,1,1,1,1)  (1,2,2,3)
We have the following tetrangle of reversed partitions:
                             0
                            (1)
                          (2)(11)
                        (3)(12)(111)
                   (4)(13)(22)(112)(1111)
             (5)(14)(23)(113)(122)(1112)(11111)
  (6)(15)(24)(114)(33)(123)(1113)(222)(1122)(11112)(111111)

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

Row lengths are A000041.
The version for reversed partitions is A026792.
The version for colex instead of revlex is A026791.
The version for lex instead of revlex is A080576.
The non-reflected version is A080577.
The number of distinct parts is A115623.
Taking Heinz numbers gives A129129.
The version for compositions is A228351.
Partition lengths are A238966.
Partition maxima are A331581.
The length-sensitive version is A334442.
Lexicographically ordered partitions are A193073.
Partitions in colexicographic order are A211992.
Cf. A036036, A036037, A112798, A129129, A228531, A296774, A334301, A334302, A334435, A334436, A334438, A334439.

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

cp's OEIS Frontend

A353432 Numbers k such that the k-th composition in standard order has its own run-lengths as a consecutive subsequence.

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A374520 Numbers k such that the leaders of maximal anti-runs in the k-th composition in standard order (A066099) are not identical.

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A124763 Number of non-rises (levels or falls) for compositions in standard order.

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A333379 Numbers k such that the k-th composition in standard order is weakly increasing and covers an initial interval of positive integers.

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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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A334266 Numbers k such that the k-th composition in standard order is both a reversed Lyndon word and a co-Lyndon word.

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A344089 Flattened tetrangle of reversed strict integer partitions, sorted first by length and then colexicographically.

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A163510 Irregular table read by rows: Write n in binary. For each 1, the m-th term of row n is the number of 0's between the m-th 1, reading right to left, and the (m-1)th 1 (or the right side of the number if m-1 = 0).

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