A114994 -id:A114994 - cp's OEIS frontend

A124769 Number of strictly decreasing runs for compositions in standard order.

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

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 the number of maximal strictly decreasing runs in this composition. Alternatively, a(n) is one plus 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. 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; the strictly increasing runs are 2,1; 1; so a(11) = 2.
The table starts:
  0
  1
  1 2
  1 1 2 3
  1 1 2 2 2 2 3 4
  1 1 1 2 2 2 2 3 2 2 3 3 3 3 4 5
  1 1 1 2 2 1 2 3 2 2 3 3 2 2 3 4 2 2 2 3 3 3 3 4 3 3 4 4 4 4 5 6

Cf. A066099, A124764, A011782 (row lengths).
Compositions of n with k weak ascents are A333213.
Positions of ones 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 (this sequence).
- 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.
Cf. A124760, A124761, A124763, A233249, A238343.

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

a(0) = 0, a(n) = A124764(n) + 1 for n > 0.

A351014 Number of distinct runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Feb 07 2022

nonn

The n-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 n, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The number 3310 has binary expansion 110011101110 and standard composition (1,3,1,1,2,1,1,2), with runs (1), (3), (1,1), (2), (1,1), (2), of which 4 are distinct, so a(3310) = 4.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Counting not necessarily distinct runs gives A124767.
Using binary expansions instead of standard compositions gives A297770.
Positions of first appearances are A351015.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A085207 represents concatenation of standard compositions, reverse A085208.
A333489 ranks anti-runs, complement A348612.
A345167 ranks alternating compositions, counted by A025047.
A351204 counts partitions where every permutation has all distinct runs.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739, ranked by A351290.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A098859, A106356, A116608, A238279, A242882, A318928, A325545, A328592, A329745, A350952, A351201.

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

A345168 Numbers k such that the k-th composition in standard order is not alternating.

Original entry on oeis.org

3, 7, 10, 11, 14, 15, 19, 21, 23, 26, 27, 28, 29, 30, 31, 35, 36, 37, 39, 42, 43, 46, 47, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 69, 71, 73, 74, 75, 78, 79, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 99, 100, 101, 103, 104, 105, 106, 107, 110

Offset: 1

Gus Wiseman, Jun 15 2021

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.

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

			The sequence of terms together with their binary indices begins:
     3: (1,1)          35: (4,1,1)        59: (1,1,2,1,1)
     7: (1,1,1)        36: (3,3)          60: (1,1,1,3)
    10: (2,2)          37: (3,2,1)        61: (1,1,1,2,1)
    11: (2,1,1)        39: (3,1,1,1)      62: (1,1,1,1,2)
    14: (1,1,2)        42: (2,2,2)        63: (1,1,1,1,1,1)
    15: (1,1,1,1)      43: (2,2,1,1)      67: (5,1,1)
    19: (3,1,1)        46: (2,1,1,2)      69: (4,2,1)
    21: (2,2,1)        47: (2,1,1,1,1)    71: (4,1,1,1)
    23: (2,1,1,1)      51: (1,3,1,1)      73: (3,3,1)
    26: (1,2,2)        52: (1,2,3)        74: (3,2,2)
    27: (1,2,1,1)      53: (1,2,2,1)      75: (3,2,1,1)
    28: (1,1,3)        55: (1,2,1,1,1)    78: (3,1,1,2)
    29: (1,1,2,1)      56: (1,1,4)        79: (3,1,1,1,1)
    30: (1,1,1,2)      57: (1,1,3,1)      83: (2,3,1,1)
    31: (1,1,1,1,1)    58: (1,1,2,2)      84: (2,2,3)

Wikipedia, Alternating permutation

The complement is A345167.
These compositions are counted by A345192.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A025047 counts alternating or wiggly compositions, directed A025048, A025049.
A344604 counts alternating compositions with twins.
A345194 counts alternating patterns (with twins: A344605).
A345164 counts alternating permutations of prime indices (with twins: A344606).
A345165 counts partitions without a alternating permutation, ranked by A345171.
A345170 counts partitions with a alternating permutation, ranked by A345172.
A348610 counts alternating ordered factorizations, complement A348613.
Statistics of standard compositions:
- Length is A000120.
- Constant runs are A124767.
- Heinz number is A333219.
- Number of maximal anti-runs is A333381.
- Runs-resistance is A333628.
- Number of distinct parts is A334028.
Classes of standard compositions:
- Weakly decreasing compositions (partitions) are A114994.
- Weakly increasing compositions (multisets) are A225620.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Anti-run compositions are A333489.
- Non-anti-run compositions are A348612.
Cf. A001222, A005649, A008965, A059893, A106356, A238279, A344615, A345162, A345163, A345166, A345169, A348377, A348380.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Select[Range[0,100],Not@*wigQ@*stc]

A333256 Numbers k such that the k-th composition in standard order is strictly decreasing.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 16, 17, 18, 32, 33, 34, 37, 64, 65, 66, 68, 69, 128, 129, 130, 132, 133, 137, 256, 257, 258, 260, 261, 264, 265, 274, 512, 513, 514, 516, 517, 520, 521, 529, 530, 549, 1024, 1025, 1026, 1028, 1029, 1032, 1033, 1040, 1041, 1042, 1058, 1061

Offset: 1

Gus Wiseman, Mar 20 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 positive terms together with the corresponding compositions begins:
     1: (1)         128: (8)         517: (7,2,1)
     2: (2)         129: (7,1)       520: (6,4)
     4: (3)         130: (6,2)       521: (6,3,1)
     5: (2,1)       132: (5,3)       529: (5,4,1)
     8: (4)         133: (5,2,1)     530: (5,3,2)
     9: (3,1)       137: (4,3,1)     549: (4,3,2,1)
    16: (5)         256: (9)        1024: (11)
    17: (4,1)       257: (8,1)      1025: (10,1)
    18: (3,2)       258: (7,2)      1026: (9,2)
    32: (6)         260: (6,3)      1028: (8,3)
    33: (5,1)       261: (6,2,1)    1029: (8,2,1)
    34: (4,2)       264: (5,4)      1032: (7,4)
    37: (3,2,1)     265: (5,3,1)    1033: (7,3,1)
    64: (7)         274: (4,3,2)    1040: (6,5)
    65: (6,1)       512: (10)       1041: (6,4,1)
    66: (5,2)       513: (9,1)      1042: (6,3,2)
    68: (4,3)       514: (8,2)      1058: (5,4,2)
    69: (4,2,1)     516: (7,3)      1061: (5,3,2,1)

Strictly increasing runs are counted by A124768.
The normal case is A246534.
The weakly decreasing version is A114994.
The weakly increasing version is A225620.
The unequal version is A233564.
The equal version is A272919.
The strictly increasing version is A333255.
Cf. A000120, A029931, A048793, A066099, A070939, A124769, A228351, A233249, A333217, A333256, A333380.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Greater@@stc[#]&]

A333255 Numbers k such that the k-th composition in standard order is strictly increasing.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 12, 16, 20, 24, 32, 40, 48, 52, 64, 72, 80, 96, 104, 128, 144, 160, 192, 200, 208, 256, 272, 288, 320, 328, 384, 400, 416, 512, 544, 576, 640, 656, 768, 784, 800, 832, 840, 1024, 1056, 1088, 1152, 1280, 1296, 1312, 1536, 1568, 1600, 1664, 1680

Offset: 1

Gus Wiseman, Mar 20 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 positive terms together with the corresponding compositions begins:
     1: (1)         128: (8)         656: (2,3,5)
     2: (2)         144: (3,5)       768: (1,9)
     4: (3)         160: (2,6)       784: (1,4,5)
     6: (1,2)       192: (1,7)       800: (1,3,6)
     8: (4)         200: (1,3,4)     832: (1,2,7)
    12: (1,3)       208: (1,2,5)     840: (1,2,3,4)
    16: (5)         256: (9)        1024: (11)
    20: (2,3)       272: (4,5)      1056: (5,6)
    24: (1,4)       288: (3,6)      1088: (4,7)
    32: (6)         320: (2,7)      1152: (3,8)
    40: (2,4)       328: (2,3,4)    1280: (2,9)
    48: (1,5)       384: (1,8)      1296: (2,4,5)
    52: (1,2,3)     400: (1,3,5)    1312: (2,3,6)
    64: (7)         416: (1,2,6)    1536: (1,10)
    72: (3,4)       512: (10)       1568: (1,4,6)
    80: (2,5)       544: (4,6)      1600: (1,3,7)
    96: (1,6)       576: (3,7)      1664: (1,2,8)
   104: (1,2,4)     640: (2,8)      1680: (1,2,3,5)

Strictly increasing runs are counted by A124768.
The normal case is A164894.
The weakly decreasing version is A114994.
The weakly increasing version is A225620.
The unequal version is A233564.
The equal version is A272919.
The strictly decreasing version is A333256.
Cf. A000120, A029931, A048793, A066099, A070939, A124762, A228351, A333217, A333220, A333379.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Less@@stc[#]&]

A333227 Numbers k such that the k-th composition in standard order is pairwise coprime, where a singleton is not coprime unless it is (1).

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 33, 35, 37, 38, 39, 41, 44, 47, 48, 49, 50, 51, 52, 55, 56, 57, 59, 60, 61, 62, 63, 65, 66, 67, 68, 71, 72, 75, 77, 78, 79, 80, 83, 89, 92, 95, 96, 97, 99, 101, 102, 103, 105

Offset: 1

Gus Wiseman, Mar 27 2020

nonn

This is the definition used for CoprimeQ in Mathematica.

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 together with the corresponding compositions begins:
   1: (1)          27: (1,2,1,1)      55: (1,2,1,1,1)
   3: (1,1)        28: (1,1,3)        56: (1,1,4)
   5: (2,1)        29: (1,1,2,1)      57: (1,1,3,1)
   6: (1,2)        30: (1,1,1,2)      59: (1,1,2,1,1)
   7: (1,1,1)      31: (1,1,1,1,1)    60: (1,1,1,3)
   9: (3,1)        33: (5,1)          61: (1,1,1,2,1)
  11: (2,1,1)      35: (4,1,1)        62: (1,1,1,1,2)
  12: (1,3)        37: (3,2,1)        63: (1,1,1,1,1,1)
  13: (1,2,1)      38: (3,1,2)        65: (6,1)
  14: (1,1,2)      39: (3,1,1,1)      66: (5,2)
  15: (1,1,1,1)    41: (2,3,1)        67: (5,1,1)
  17: (4,1)        44: (2,1,3)        68: (4,3)
  18: (3,2)        47: (2,1,1,1,1)    71: (4,1,1,1)
  19: (3,1,1)      48: (1,5)          72: (3,4)
  20: (2,3)        49: (1,4,1)        75: (3,2,1,1)
  23: (2,1,1,1)    50: (1,3,2)        77: (3,1,2,1)
  24: (1,4)        51: (1,3,1,1)      78: (3,1,1,2)
  25: (1,3,1)      52: (1,2,3)        79: (3,1,1,1,1)

A different ranking of the same compositions is A326675.
Ignoring repeated parts gives A333228.
Let q(k) be the k-th composition in standard order:
- The terms of q(k) are row k of A066099.
- The sum of q(k) is A070939(k).
- The product of q(k) is A124758(k).
- q(k) has A124767(k) runs and A333381(k) anti-runs.
- The GCD of q(k) is A326674(k).
- The Heinz number of q(k) is A333219(k).
- The LCM of q(k) is A333226(k).
Coprime or singleton sets are ranked by A087087.
Strict compositions are ranked by A233564.
Constant compositions are ranked by A272919.
Relatively prime compositions appear to be ranked by A291166.
Normal compositions are ranked by A333217.
Cf. A000120, A029931, A048793, A096111, A114994, A225620, A228351.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,120],CoprimeQ@@stc[#]&]

A124771 Number of distinct subsequences for compositions in standard order.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.
From Vladimir Shevelev, Dec 18 2013: (Start)
Every number in binary is a concatenation of parts of the form 10...0 with k>=0 zeros. For example, 5=(10)(1), 11=(10)(1)(1), 7=(1)(1)(1). We call d>0 a c-divisor of m, if d consists of some consecutive parts of m taking from the left to the right. Note that, to d=0 corresponds an empty set of parts. So it is natural to consider 0 as a c-divisor of every m. For example, 5=(10)(1) is a divisor of 23=(10)(1)(1)(1). Analogously, 1,2,3,7,11,23 are c-divisors of 23. But 6=(1)(10) is not a c-divisor of 23.
One can prove a one-to-one correspondence between distinct subsequences for  composition no. n in standard order and c-divisors of n. So, the sequence lists also numbers of c-divisors of nonnegative integers.
(End)
These are contiguous subsequences, or restrictions to a subinterval. The case for all subsequences is A334299. - Gus Wiseman, Jun 02 2020

			Composition number 11 is 2,1,1; the subsequences are (empty); 1; 2; 1,1; 2,1; 2,1,1; so a(11) = 6.
The table starts:
1
2
1 2
1 3 3 3
Let n=11=(10)(1)(1). We have the following c-divisors of 11: 0,1,2,3,5,11. Thus a(11)=6. Note, that 3=(1)(1) is not a c-divisor of 13=(1)(10)(1) since, although it contains parts of 3=(1)(1), but in non-consecutive order. The c-divisors of 13 are 0,1,2,5,6,13. So, a(13)=6.
From _Gus Wiseman_, Jun 01 2020: (Start)
The c-divisors of n are given in column n below:
  0  0  0  0  0  0  0  0  0  0  0   0   0   0   0   0   0   0   0
     1  2  1  4  1  1  1  8  1  2   1   1   1   1   1   16  1   2
           3     2  2  3     4  10  2   4   2   2   3       8   4
                 5  6  7     9      3   12  5   3   7       17  18
                                    5       6   6   15
                                    11      13  14
(End)

Alois P. Heinz, Rows n = 0..14, flattened

Cf. A000005, A011782 (row lengths), A066099, A114994, A233249, A233312.
Not allowing empty subsequences gives A124770.
Dominates A333257.
The case for not just contiguous subsequences is A334299.
Positions of first appearances are A335279.
Compositions where every subinterval has a different sum are A333222.
Knapsack compositions are A333223.
Cf. A000120, A003022, A029931, A070939, A108917, A124767, A325680, A325770, A333224, A334967, A334968.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n],{_,s___,_}:>{s}]]],{n,0,100}] (* Gus Wiseman, Jun 01 2020 *)

a(n) = A124770(n) + 1.
From Vladimir Shevelev, Dec 18 2013: (Start)
a(2^n) = 2. Note that in concatenation representations of integers in binary, numbers {2^k}, k>=0, play the role of primes. So the formula is an analog of A000005(prime(n))=2.
a(2^n-1) = n+1; for n>=2, a(2^n+1) = 4.
For c-equivalent numbers n_1 and n_2 (i.e., differed only by order of parts) we have a(n_1) = a(n_2). For example, a(24)=a(17)=4. If the canonical representation of n is n=(1)^k_1[*](10)^k_2[*](100)^k_3[*]... , where [*] denotes operation of concatenation (cf. A233569), then a(n)<=(k_1+1)*(k_2+1)*...
(End)

A333257 Number of distinct consecutive subsequence-sums of the k-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Mar 20 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 ninth composition in standard order is (3,1), which has consecutive subsequences (), (1), (3), (3,1), with sums 0, 1, 3, 4, so a(9) = 4.

Dominated by A124771.
Compositions where every subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222, while the case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack partitions are counted by A108917 and ranked by A299702.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223.
The version for Heinz numbers of partitions is A325770.
Not allowing empty subsequences gives A333224.
Cf. A000120, A029931, A048793, A059519, A066099, A070939, A114994, A124765, A124767, A233564, A272919, A325778, A333217.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Union[ReplaceList[stc[n],{_,s___,_}:>Plus[s]]]],{n,0,100}]

a(n) = A333224(n) + 1.

A333766 Maximum part of the n-th composition in standard order. a(0) = 0.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 05 2020

nonn

One plus the longest run of 0's in the binary expansion of n.

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. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The 100th composition in standard order is (1,3,3), so a(100) = 3.

Positions of ones are A000225.
Positions of terms <= 2 are A003754.
The version for prime indices is A061395.
Positions of terms > 1 are A062289.
Positions of first appearances are A131577.
The minimum part is given by A333768.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Compositions without 1's are A022340.
- Sum is A070939.
- Product is A124758.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Runs-resistance is A333628.
- Weakly decreasing compositions are A114994.
- Weakly increasing compositions are A225620.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
Cf. A029931, A048793, A087117, A228351, A328594, A333217, A333218, A333219, A333632, A333767.

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

For n > 0, a(n) = A087117(n) + 1.

A353849 Number of distinct positive run-sums of the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 30 2022

nonn

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

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.

			Composition 462903 in standard order is (1,1,4,7,1,2,1,1,1), with run-sums (2,4,7,1,2,3), of which a(462903) = 5 are distinct.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Counting repeated runs also gives A124767.
Positions of first appearances are A246534.
For distinct runs instead of run-sums we have A351014 (firsts A351015).
A version for partitions is A353835, weak A353861.
Positions of 1's are A353848, counted by A353851.
The version for binary expansion is A353929 (firsts A353930).
The run-sums themselves are listed by A353932, with A353849 distinct terms.
For distinct run-lengths instead of run-sums we have A354579.
A005811 counts runs in binary expansion.
A066099 lists compositions in standard order.
A165413 counts distinct run-lengths in binary expansion.
A297770 counts distinct runs in binary expansion, firsts A350952.
A353847 represents the run-sum transformation for compositions.
A353853-A353859 pertain to composition run-sum trajectory.
Distinct runs: A032020, A175413, A351013, A351018, A329739, A351290.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A003242, A044813, A071625, A238279, A329738, A333381, A333489, A333755, A353744, A353832, A353850, A353852, A353866.

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

cp's OEIS Frontend

A124769 Number of strictly decreasing runs for compositions in standard order.

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A351014 Number of distinct runs in the n-th composition in standard order.

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A345168 Numbers k such that the k-th composition in standard order is not alternating.

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A333256 Numbers k such that the k-th composition in standard order is strictly decreasing.

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A333255 Numbers k such that the k-th composition in standard order is strictly increasing.

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A333227 Numbers k such that the k-th composition in standard order is pairwise coprime, where a singleton is not coprime unless it is (1).

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A124771 Number of distinct subsequences for compositions in standard order.

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A333257 Number of distinct consecutive subsequence-sums of the k-th composition in standard order.

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A333766 Maximum part of the n-th composition in standard order. a(0) = 0.