A228351 -id:A228351 - cp's OEIS frontend

A124761 Number of falls for compositions in standard order.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 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 weakly increasing runs in this composition. Alternatively, a(n) is the number of strict descents in the same composition. For example, the weakly increasing of runs of the 1234567th composition are ((3),(2),(1,2,2),(1,2,5),(1,1,1)), so a(1234567) = 5 - 1 = 4. The 4 strict descents together with the weak 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) = 1.
The table starts:
  0
  0
  0 0
  0 1 0 0
  0 1 0 1 0 1 0 0
  0 1 1 1 0 1 1 1 0 1 0 1 0 1 0 0
  0 1 1 1 0 2 1 1 0 1 0 1 1 2 1 1 0 1 1 1 0 1 1 1 0 1 0 1 0 1 0 0

Cf. A066099, A124760, A124763, A124764, A011782 (row lengths), A045883 (row sums), A333213, A333220, A333379.
Positions of zeros are A225620.
Compositions of n with k strict descents are A238343.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- 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.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Runs-resistance is A333628.

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

For a composition b(1),...,b(k), a(n) = Sum_{1<=i=1b(i+1)} 1.

For n > 0, a(n) = A124766(n) - 1. - Gus Wiseman, Apr 08 2020

A335239 Numbers k such that the k-th composition in standard-order (A066099) does not have all pairwise coprime parts, where a singleton is not coprime unless it is (1).

Original entry on oeis.org

0, 2, 4, 8, 10, 16, 21, 22, 26, 32, 34, 36, 40, 42, 43, 45, 46, 53, 54, 58, 64, 69, 70, 73, 74, 76, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 98, 100, 104, 106, 107, 109, 110, 117, 118, 122, 128, 130, 136, 138, 139, 141, 142, 146, 147, 148, 149, 150, 153

Offset: 1

Gus Wiseman, May 28 2020

nonn

We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).

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 the corresponding compositions begins:
    0: ()            45: (2,1,2,1)     86: (2,2,1,2)
    2: (2)           46: (2,1,1,2)     87: (2,2,1,1,1)
    4: (3)           53: (1,2,2,1)     88: (2,1,4)
    8: (4)           54: (1,2,1,2)     90: (2,1,2,2)
   10: (2,2)         58: (1,1,2,2)     91: (2,1,2,1,1)
   16: (5)           64: (7)           93: (2,1,1,2,1)
   21: (2,2,1)       69: (4,2,1)       94: (2,1,1,1,2)
   22: (2,1,2)       70: (4,1,2)       98: (1,4,2)
   26: (1,2,2)       73: (3,3,1)      100: (1,3,3)
   32: (6)           74: (3,2,2)      104: (1,2,4)
   34: (4,2)         76: (3,1,3)      106: (1,2,2,2)
   36: (3,3)         81: (2,4,1)      107: (1,2,2,1,1)
   40: (2,4)         82: (2,3,2)      109: (1,2,1,2,1)
   42: (2,2,2)       84: (2,2,3)      110: (1,2,1,1,2)
   43: (2,2,1,1)     85: (2,2,2,1)    117: (1,1,2,2,1)

The complement is A333227.
The version without singletons is A335236.
Ignoring repeated parts gives A335238.
Singleton or pairwise coprime partitions are counted by A051424.
Singleton or pairwise coprime sets are ranked by A087087.
Numbers whose binary indices are pairwise coprime are A326675.
Coprime partitions are counted by A327516.
Non-coprime partitions are counted by A335240.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Product is A124758.
- Reverse is A228351
- GCD is A326674.
- Heinz number is A333219.
- LCM is A333226.
- Coprime compositions are A333227.
- Compositions whose distinct parts are coprime are A333228.
- Number of distinct parts is A334028.
Cf. A007360, A048793, A101268, A291166, A302569, A335235, A335237.

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

A375137 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 1-32.

Original entry on oeis.org

50, 98, 101, 114, 178, 194, 196, 197, 202, 203, 210, 226, 229, 242, 306, 324, 354, 357, 370, 386, 388, 389, 393, 394, 395, 402, 404, 405, 406, 407, 418, 421, 434, 450, 452, 453, 458, 459, 466, 482, 485, 498, 562, 610, 613, 626, 644, 649, 690, 706, 708, 709

Offset: 1

Gus Wiseman, Aug 09 2024

nonn

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.
These are also numbers k such that the maximal weakly increasing runs in the k-th composition in standard order do not have weakly decreasing leaders, where the leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
The reverse version (A375138) ranks compositions matching the dashed pattern 23-1.

			Composition 102 is (1,3,1,2), which matches 1-3-2 but not 1-32.
Composition 210 is (1,2,3,2), which matches 1-32 but not 132.
Composition 358 is (2,1,3,1,2), which matches 2-3-1 and 1-3-2 but not 23-1 or 1-32.
The terms together with corresponding compositions begin:
   50: (1,3,2)
   98: (1,4,2)
  101: (1,3,2,1)
  114: (1,1,3,2)
  178: (2,1,3,2)
  194: (1,5,2)
  196: (1,4,3)
  197: (1,4,2,1)
  202: (1,3,2,2)
  203: (1,3,2,1,1)
  210: (1,2,3,2)
  226: (1,1,4,2)
  229: (1,1,3,2,1)
  242: (1,1,1,3,2)

Wikipedia, Permutation pattern.

The complement is too dense, but counted by A189076.
The non-dashed version is A335480, reverse A335482.
For leaders of identical runs we have A335485, reverse A335486.
For identical leaders we have A374633, counted by A374631.
Compositions of this type are counted by A374636.
For distinct leaders we have A374768, counted by A374632.
The reverse version is A375138, counted by A374636.
For leaders of strictly increasing runs we have A375139, counted by A375135.
Matching 1-21 also gives A375295, counted by A375140 (complement A188920).
A003242 counts anti-runs, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Leader is A065120.
- Parts are listed by A066099, reverse A228351.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Run-length transform is A333627, sum A070939.
- Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
Cf. A056823, A106356, A188919, A238343, A333213, A373948, A373953, A374634, A374635, A374637, A375123, A375296.

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

A124764 Number of non-falls (levels or rises) for compositions in standard order.

Original entry on oeis.org

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

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 decreasing runs in this composition. Alternatively, a(n) is the number of weak ascents in the same composition. For example, the strictly decreasing runs of the 1234567th composition are ((3,2,1),(2),(2,1),(2),(5,1),(1),(1)), so a(1234567) = 7 - 1 = 6. The 6 weak ascents together with the strict descents 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) = 1.
The table starts:
  0
  0
  0 1
  0 0 1 2
  0 0 1 1 1 1 2 3
  0 0 0 1 1 1 1 2 1 1 2 2 2 2 3 4
  0 0 0 1 1 0 1 2 1 1 2 2 1 1 2 3 1 1 1 2 2 2 2 3 2 2 3 3 3 3 4 5

Cf. A066099, A124760, A124761, A124762, A124763, A011782 (row lengths), A045883 (row sums), A233249, A238343.
Compositions of n with k weak ascents are A333213.
Positions of zeros are A333256.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Partial sums from the right are A048793 (triangle).
- Sum is A070939.
- Weakly decreasing compositions are A114994.
- 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.
- Reversed initial intervals A164894.
- Weakly increasing compositions are A225620.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- 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],LessEqual@@#&]],{n,0,100}] (* Gus Wiseman, Apr 08 2020 *)

a(n) = A124760(n) + A124762(n)
For a composition b(1),...,b(k), a(n) = Sum_{1<=i=1=b(i+1)} 1.
For n > 0, a(n) = A124769(n) - 1. - Gus Wiseman, Apr 08 2020

A334265 Numbers k such that the k-th composition in standard order is a reversed 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, 41, 43, 47, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 87, 91, 95, 128, 129, 130, 131, 132, 133, 135, 137, 138, 139, 141, 143, 145, 146, 147, 149, 151, 155, 159, 161, 163

Offset: 1

Gus Wiseman, Apr 22 2020

nonn

Reversed Lyndon words are different from co-Lyndon words (A326774).
A Lyndon word is a finite sequence of positive integers that is lexicographically strictly less than all of its cyclic rotations.
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 words begins:
    0: ()            37: (3,2,1)         83: (2,3,1,1)
    1: (1)           39: (3,1,1,1)       85: (2,2,2,1)
    2: (2)           41: (2,3,1)         87: (2,2,1,1,1)
    4: (3)           43: (2,2,1,1)       91: (2,1,2,1,1)
    5: (2,1)         47: (2,1,1,1,1)     95: (2,1,1,1,1,1)
    8: (4)           64: (7)            128: (8)
    9: (3,1)         65: (6,1)          129: (7,1)
   11: (2,1,1)       66: (5,2)          130: (6,2)
   16: (5)           67: (5,1,1)        131: (6,1,1)
   17: (4,1)         68: (4,3)          132: (5,3)
   18: (3,2)         69: (4,2,1)        133: (5,2,1)
   19: (3,1,1)       71: (4,1,1,1)      135: (5,1,1,1)
   21: (2,2,1)       73: (3,3,1)        137: (4,3,1)
   23: (2,1,1,1)     74: (3,2,2)        138: (4,2,2)
   32: (6)           75: (3,2,1,1)      139: (4,2,1,1)
   33: (5,1)         77: (3,1,2,1)      141: (4,1,2,1)
   34: (4,2)         79: (3,1,1,1,1)    143: (4,1,1,1,1)
   35: (4,1,1)       81: (2,4,1)        145: (3,4,1)

The non-reversed version is A275692.
The generalization to necklaces is A333943.
The dual version (reversed co-Lyndon words) is A328596.
The case that is also co-Lyndon is A334266.
Binary Lyndon words are counted by A001037.
Lyndon compositions are counted by A059966.
Normal Lyndon words are counted by A060223.
Numbers whose prime signature is a reversed Lyndon word are A334298.
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 (this sequence).
- Co-Lyndon words are A326774.
- Reversed co-Lyndon words are A328596.
- Length of Lyndon factorization is A329312.
- Distinct rotations are counted by A333632.
- Lyndon factorizations are counted by A333940.
- Length of Lyndon factorization of reverse is A334297.
Cf. A000031, A027375, A138904, A211100, A328595, A329131, A329313, A333764, A334028, A334267.

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];
Select[Range[0,100],lynQ[Reverse[stc[#]]]&]

A335404 Numbers k such that the k-th composition in standard order (A066099) has the same product as sum.

Original entry on oeis.org

1, 2, 4, 8, 10, 16, 32, 37, 38, 41, 44, 50, 52, 64, 128, 139, 141, 142, 163, 171, 173, 174, 177, 181, 182, 184, 186, 197, 198, 209, 213, 214, 216, 218, 226, 232, 234, 256, 295, 307, 313, 316, 403, 409, 412, 457, 460, 484, 512, 535, 539, 541, 542, 647, 707, 737

Offset: 1

Gus Wiseman, Jun 06 2020

nonn

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 the corresponding compositions begins:
    1: (1)
    2: (2)
    4: (3)
    8: (4)
   10: (2,2)
   16: (5)
   32: (6)
   37: (3,2,1)
   38: (3,1,2)
   41: (2,3,1)
   44: (2,1,3)
   50: (1,3,2)
   52: (1,2,3)
   64: (7)
  128: (8)
  139: (4,2,1,1)
  141: (4,1,2,1)
  142: (4,1,1,2)
  163: (2,4,1,1)
  171: (2,2,2,1,1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

The lengths of standard compositions are given by A000120.
Sum of binary indices is A029931.
Sum of prime indices is A056239.
Sum of standard compositions is A070939.
Product of standard compositions is A124758.
Taking GCD instead of product gives A131577.
The version for prime indices is A301987.
The version for prime indices of nonprime numbers is A301988.
These compositions are counted by A335405.
Cf. A001055, A003963, A066099, A096111, A124767, A228351, A233249, A272919, A333219, A333220, A331579.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],Times@@stc[#]==Plus@@stc[#]&]

A124758(a(n)) = A070939(a(n)).

A344086 Flattened tetrangle of strict integer partitions sorted first by sum, then lexicographically.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 11 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (21)(3)
  4: (31)(4)
  5: (32)(41)(5)
  6: (321)(42)(51)(6)
  7: (421)(43)(52)(61)(7)
  8: (431)(521)(53)(62)(71)(8)
  9: (432)(531)(54)(621)(63)(72)(81)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
Taking revlex instead of lex gives A118457.
The not necessarily strict version is A193073.
The version for reversed partitions is A246688.
The Heinz numbers of these partitions grouped by sum are A246867.
The ordered generalization is A339351.
Taking colex instead of lex gives A344087.
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts reversed strict partitions by Heinz number.
A329631 sorts strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A211992, A228100, A228351, A228531, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A344085, A344086, A344088, A344089.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Table[Sort[Select[IntegerPartitions[n],UnsameQ@@#&],lexsort],{n,0,8}]

A353402 Numbers k such that the k-th composition in standard order has its own run-lengths as a subsequence (not necessarily consecutive).

Original entry on oeis.org

0, 1, 10, 21, 26, 43, 53, 58, 107, 117, 174, 186, 292, 314, 346, 348, 349, 373, 430, 442, 570, 585, 586, 629, 676, 693, 696, 697, 698, 699, 804, 826, 858, 860, 861, 885, 954, 1082, 1141, 1173, 1210, 1338, 1353, 1387, 1392, 1393, 1394, 1396, 1397, 1398, 1466

Offset: 0

Gus Wiseman, May 15 2022

nonn

First differs from A353432 (the consecutive case) in having 0 and 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)
   53:     110101  (1,2,2,1)
   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)

The version for partitions is A325755, counted by A325702.
These compositions are counted by A353390.
The recursive version is A353431, counted by A353391.
The consecutive case is A353432, counted by A353392.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, reverse A228351.
A333769 lists run-lengths of compositions in standard order.
Words with all distinct run-lengths: A032020, A044813, A098859, A130091, A329739, A351017.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767, distinct A351014.
- Subsequences are counted by A334299, consecutive A124770/A124771.
- Runs-resistance is A333628.
Classes of standard compositions:
- Partitions are A114994, strict A333255, rev A225620, strict rev A333256.
- Runs are A272919.
- Golomb rulers are A333222, counted by A169942.
- Knapsack compositions are A333223, counted by A325676.
- Anti-runs are A333489, counted by A003242.
Cf. A114640, A165413, A181819, A318928, A325705, A329738, A333224/A333257, A333755, A353393, A353403, A353430.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
rosQ[y_]:=Length[y]==0||MemberQ[Subsets[y],Length/@Split[y]];
Select[Range[0,100],rosQ[stc[#]]&]

A228370 Toothpick sequence from a diagram of compositions of the positive integers (see Comments lines for definition).

Original entry on oeis.org

0, 1, 2, 4, 6, 7, 8, 11, 15, 16, 17, 19, 21, 22, 23, 27, 35, 36, 37, 39, 41, 42, 43, 46, 50, 51, 52, 54, 56, 57, 58, 63, 79, 80, 81, 83, 85, 86, 87, 90, 94, 95, 96, 98, 100, 101, 102, 106, 114, 115, 116, 118, 120, 121, 122, 125, 129, 130, 131, 133, 135, 136, 137, 143, 175

Offset: 0

Omar E. Pol, Aug 21 2013

nonn

In order to construct this sequence we use the following rules:
We start in the first quadrant of the square grid with no toothpicks, so a(0) = 0.
If n is odd then at stage n we place the smallest possible number of toothpicks of length 1 connected by their endpoints in horizontal direction starting from the grid point (0, (n+1)/2) such that the x-coordinate of the exposed endpoint of the last toothpick is not equal to the x-coordinate of any outer corner of the structure.
If n is even then at stage n we place toothpicks of length 1 connected by their endpoints in vertical direction, starting from the exposed toothpick endpoint, downward up to touch the structure or up to touch the x-axis.
Note that the number of toothpick of added at stage (n+1)/2 in horizontal direction is also A001511(n) and the number of toothpicks added at stage n/2 in vertical direction is also A006519(n).
The sequence gives the number of toothpicks after n stages. A228371 (the first differences) gives the number of toothpicks added at the n-th stage.
After 2^k stages a new section of the structure is completed, so the structure can be interpreted as a diagram of the 2^(k-1) compositions of k in colexicographic order, if k >= 1 (see A228525). The infinite diagram can be interpreted as a table of compositions of the positive integers.
The equivalent sequence for partitions is A225600.

			For n = 32 the diagram represents the 16 compositions of 5. The structure has 79 toothpicks, so a(32) = 79. Note that the k-th horizontal line segment has length A001511(k) equals the largest part of the k-th region, and the k-th vertical line segment has length A006519(k) equals the number of parts of the k-th region.
----------------------------------------------------------
.                                    Triangle
Compositions                  of compositions (rows)
of 5          Diagram          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
.
Illustration of initial terms (n = 1..16):
.
.                                   _        _
.                   _ _    _ _      _ _      _|_
.       _     _     _      _  |     _  |     _  |
.              |     |      | |      | |      | |
.
.       1      2     4      6        7        8
.
.
.                                            _ _
.                        _         _         _
.     _ _ _    _ _ _     _ _ _     _|_ _     _|_ _
.     _        _    |    _    |    _    |    _    |
.     _|_      _|_  |    _|_  |    _|_  |    _|_  |
.     _  |     _  | |    _  | |    _  | |    _  | |
.      | |      | | |     | | |     | | |     | | |
.
.       11       15        16        17        19
.
.
.                                _ _ _ _    _ _ _ _
.             _        _         _          _      |
.    _ _      _ _      _|_       _|_        _|_    |
.    _  |     _  |     _  |      _  |       _  |   |
.    _|_|_    _|_|_    _|_|_     _|_|_      _|_|_  |
.    _    |   _    |   _    |    _    |     _    | |
.    _|_  |   _|_  |   _|_  |    _|_  |     _|_  | |
.    _  | |   _  | |   _  | |    _  | |     _  | | |
.     | | |    | | |    | | |     | | |      | | | |
.
.      21       22       23        27          35
.

Cf. A001511, A005187, A006519, A006520, A011782, A139250, A187816, A187818, A206437, A225600, A228350, A228351, A228366, A228367, A228368, A228371, A228525, A228526.

def A228370(n): return sum(((m:=(i>>1)+1)&-m).bit_length() if i&1 else (m:=i>>1)&-m for i in range(1,n+1)) # Chai Wah Wu, Jul 14 2022

a(n) = sum_{k=1..n} A228371(k), n >= 1.
a(2n-1) = A005187(n) + A006520(n+1) - A006519(n), n >= 1.
a(2n) = A005187(n) + A006520(n+1), n >= 1.

A353431 Numbers k such that the k-th composition in standard order is empty, a singleton, or has its own run-lengths as a subsequence (not necessarily consecutive) that is already counted.

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 16, 32, 43, 58, 64, 128, 256, 292, 349, 442, 512, 586, 676, 697, 826, 1024, 1210, 1338, 1393, 1394, 1396, 1594, 2048, 2186, 2234, 2618, 2696, 2785, 2786, 2792, 3130, 4096, 4282, 4410, 4666, 5178, 5569, 5570, 5572, 5576, 5584, 6202, 8192

Offset: 1

Gus Wiseman, May 16 2022

nonn

First differs from A353696 (the consecutive version) in having 22318, corresponding to the binary word 101011100101110 and standard composition (2,2,1,1,3,2,1,1,2), whose run-lengths (2,2,1,1,2,1) are subsequence but not a consecutive subsequence.

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)
     2:          10  (2)
     4:         100  (3)
     8:        1000  (4)
    10:        1010  (2,2)
    16:       10000  (5)
    32:      100000  (6)
    43:      101011  (2,2,1,1)
    58:      111010  (1,1,2,2)
    64:     1000000  (7)
   128:    10000000  (8)
   256:   100000000  (9)
   292:   100100100  (3,3,3)
   349:   101011101  (2,2,1,1,2,1)
   442:   110111010  (1,2,1,1,2,2)
   512:  1000000000  (10)
   586:  1001001010  (3,3,2,2)
   676:  1010100100  (2,2,3,3)
   697:  1010111001  (2,2,1,1,3,1)

The non-recursive version for partitions is A325755, counted by A325702.
These compositions are counted by A353391.
The version for partitions A353393, counted by A353426, w/o primes A353389.
The non-recursive version is A353402, counted by A353390.
The non-recursive consecutive case is A353432, counted by A353392.
The consecutive case 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, multisets A225620, strict A333255, sets A333256.
- Constant compositions are A272919, counted by A000005.
- Golomb rulers are A333222, counted by A169942.
- Knapsack compositions are A333223, counted by A325676.
- Anti-runs are A333489, counted by A003242.
Cf. A032020, A044813, A114640, A165413, A181819, A329739, A318928, A325705, A333224, A353427, A353403.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
rorQ[y_]:=Length[y]<=1||MemberQ[Subsets[y],Length/@Split[y]]&& rorQ[Length/@Split[y]];
Select[Range[0,100],rorQ[stc[#]]&]

cp's OEIS Frontend

A124761 Number of falls for compositions in standard order.

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A335239 Numbers k such that the k-th composition in standard-order (A066099) does not have all pairwise coprime parts, where a singleton is not coprime unless it is (1).

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A375137 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 1-32.

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A124764 Number of non-falls (levels or rises) for compositions in standard order.

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A334265 Numbers k such that the k-th composition in standard order is a reversed Lyndon word.

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A335404 Numbers k such that the k-th composition in standard order (A066099) has the same product as sum.

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A344086 Flattened tetrangle of strict integer partitions sorted first by sum, then lexicographically.

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A353402 Numbers k such that the k-th composition in standard order has its own run-lengths as a subsequence (not necessarily consecutive).

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A228370 Toothpick sequence from a diagram of compositions of the positive integers (see Comments lines for definition).