A124771 -id:A124771 - cp's OEIS frontend

A335279 Positions of first appearances in A124771 = number of distinct contiguous subsequences of compositions in standard order.

0, 1, 3, 5, 11, 15, 23, 27, 37, 47, 55, 107, 111, 119, 155, 215, 223, 239, 411, 431, 471, 479, 495, 549, 631, 943, 951, 959, 991, 1647, 1887, 1967, 1983, 2015, 2543, 2935, 3703, 3807, 3935, 3967, 4031, 6639, 6895, 7407, 7871, 7903, 8063, 8127, 10207, 13279

Offset: 1

Gus Wiseman, Jun 03 2020

nonn

			The sequence together with the corresponding compositions begins:
     0: ()                215: (1,2,2,1,1,1)
     1: (1)               223: (1,2,1,1,1,1,1)
     3: (1,1)             239: (1,1,2,1,1,1,1)
     5: (2,1)             411: (1,3,1,2,1,1)
    11: (2,1,1)           431: (1,2,2,1,1,1,1)
    15: (1,1,1,1)         471: (1,1,2,2,1,1,1)
    23: (2,1,1,1)         479: (1,1,2,1,1,1,1,1)
    27: (1,2,1,1)         495: (1,1,1,2,1,1,1,1)
    37: (3,2,1)           549: (4,3,2,1)
    47: (2,1,1,1,1)       631: (3,1,1,2,1,1,1)
    55: (1,2,1,1,1)       943: (1,1,2,2,1,1,1,1)
   107: (1,2,2,1,1)       951: (1,1,2,1,2,1,1,1)
   111: (1,2,1,1,1,1)     959: (1,1,2,1,1,1,1,1,1)
   119: (1,1,2,1,1,1)     991: (1,1,1,2,1,1,1,1,1)
   155: (3,1,2,1,1)      1647: (1,3,1,2,1,1,1,1)
The subsequences for n = 0, 1, 3, 5, 11, 15, 23, 27 are the following (0 = empty partition):
  0  0  0   0   0    0     0     0     0    0
     1  1   1   1    1     1     1     1    1
        11  2   2    11    2     2     2    2
            21  11   111   11    11    3    11
                21   1111  21    12    21   21
                211        111   21    32   111
                           211   121   321  211
                           2111  211        1111
                                 1211       2111
                                            21111

Positions of first appearances in A124771.
Compositions where every subinterval has a different sum are A333222.
Knapsack compositions are A333223.
Cf. A000120, A003022, A029931, A066099, A070939, A124767, A124770, A325770, A334299, A334968.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
seq=Table[Length[Union[ReplaceList[stc[n],{_,s___,_}:>{s}]]],{n,0,1000}];
Table[Position[seq,i][[1,1]]-1,{i,First/@Gather[seq]}]

A233564 c-squarefree numbers: positive integers which in binary are concatenation of distinct parts of the form 10...0 with nonnegative number of zeros.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 12, 16, 17, 18, 20, 24, 32, 33, 34, 37, 38, 40, 41, 44, 48, 50, 52, 64, 65, 66, 68, 69, 70, 72, 80, 81, 88, 96, 98, 104, 128, 129, 130, 132, 133, 134, 137, 140, 144, 145, 152, 160, 161, 176, 192, 194, 196, 200, 208, 256, 257, 258, 260, 261

Offset: 1

Vladimir Shevelev, Dec 13 2013

Number of terms in interval [2^(n-1), 2^n) is the number of compositions of n with distinct parts (cf. A032020). For example, if n=6, then interval [2^5, 2^6) contains  11 terms {32,...,52}. This corresponds to 11 compositions with distinct parts of 6: 6, 5+1, 1+5, 4+2, 2+4, 3+2+1, 3+1+2, 2+3+1, 2+1+3, 1+3+2, 1+2+3.
From Gus Wiseman, Apr 06 2020: (Start)
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. This sequence lists all numbers k such that the k-th composition in standard order is strict. For example, the sequence together with the corresponding strict compositions begins:
()          38: (3,1,2)     98: (1,4,2)
(1)         40: (2,4)      104: (1,2,4)
(2)         41: (2,3,1)    128: (8)
(3)         44: (2,1,3)    129: (7,1)
(2,1)       48: (1,5)      130: (6,2)
(1,2)       50: (1,3,2)    132: (5,3)
(4)         52: (1,2,3)    133: (5,2,1)
(3,1)       64: (7)        134: (5,1,2)
(1,3)       65: (6,1)      137: (4,3,1)
(5)         66: (5,2)      140: (4,1,3)
(4,1)       68: (4,3)      144: (3,5)
(3,2)       69: (4,2,1)    145: (3,4,1)
(2,3)       70: (4,1,2)    152: (3,1,4)
(1,4)       72: (3,4)      160: (2,6)
(6)         80: (2,5)      161: (2,5,1)
(5,1)       81: (2,4,1)    176: (2,1,5)
(4,2)       88: (2,1,4)    192: (1,7)
(3,2,1)     96: (1,6)      194: (1,5,2)
(End)

			49 in binary has the following parts of the form 10...0 with nonnegative number of  zeros: (1),(1000),(1). Two of them are the same. So it is not in the sequence. On the other hand, 50 has distinct parts (1)(100)(10), thus it is a term.

Index entries for sequences related to binary expansion of n

Cf. A032020, A124771, A233249, A233312, A233416, A233420, A233569, A233655.
A subset of A333489 and superset of A333218.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Weighted sum is A029931.
- Partial sums from the right are A048793.
- Sum is A070939.
- Runs are counted by A124767.
- Reversed initial intervals A164894.
- Initial intervals are A246534.
- Constant compositions are A272919.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Anti-runs are counted by A333381.
- Anti-runs are A333489.
Cf. A114994, 225620, A228351, A238279, A242882, A329739, A329744, A333217.

bitPatt[n_]:=bitPatt[n]=Split[IntegerDigits[n,2],#1>#2||#2==0&];
Select[Range[0,300],bitPatt[#]==DeleteDuplicates[bitPatt[#]]&] (* Peter J. C. Moses, Dec 13 2013 *)
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@stc[#]&] (* Gus Wiseman, Apr 04 2020 *)

More terms from Peter J. C. Moses, Dec 13 2013

0 prepended by Gus Wiseman, Apr 04 2020

A335456 Number of normal patterns matched by compositions of n.

Original entry on oeis.org

1, 2, 5, 12, 32, 84, 211, 556, 1446, 3750, 9824, 25837, 67681, 178160, 468941, 1233837, 3248788, 8554709

Offset: 0

Gus Wiseman, Jun 16 2020

A composition of n is a finite sequence of positive integers summing to n.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The 8 compositions of 4 together with the a(4) = 32 patterns they match:
  4:   31:   13:   22:   211:   121:   112:   1111:
-----------------------------------------------------
  ()   ()    ()    ()    ()     ()     ()     ()
  (1)  (1)   (1)   (1)   (1)    (1)    (1)    (1)
       (21)  (12)  (11)  (11)   (11)   (11)   (11)
                         (21)   (12)   (12)   (111)
                         (211)  (21)   (112)  (1111)
                                (121)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

References found in the link are not all included here.
The version for standard compositions is A335454.
The contiguous case is A335457.
The version for Heinz numbers of partitions is A335549.
Patterns are counted by A000670 and ranked by A333217.
The n-th composition has A124771(n) distinct consecutive subsequences.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A333257(n) distinct subsequence-sums.
The n-th composition has A334299(n) distinct subsequences.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A106356, A108917, A124770, A269134, A329744, A333224, A335458.

mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Sum[Length[Union[mstype/@Subsets[y]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,0,8}]

a(14)-a(16) from Jinyuan Wang, Jun 26 2020

a(17) from John Tyler Rascoe, Mar 14 2025

A335465 Number of minimal normal patterns avoided by the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 12, 4, 3, 3, 3, 3, 4, 3, 4, 12, 4, 3, 12, 4, 12, 4, 12, 4, 3, 3, 3, 3, 4, 3, 3, 6, 4, 3, 6, 3, 3, 6, 10, 10, 4, 3, 12, 6, 12, 3, 10, 10, 12, 4, 12, 3, 12, 4, 12, 4, 3, 3, 3, 3, 4, 3, 3, 6

Offset: 0

Gus Wiseman, Jun 20 2020

These patterns comprise the basis of the class of patterns generated by this composition.
We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).
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 bases of classes generated by (), (1), (2,1,1), (3,1,2), (2,1,2,1), and (1,2,1), corresponding to n = 0, 1, 11, 38, 45, 13, are the respective columns below.
  (1)  (1,1)  (1,2)    (1,1)    (1,1,1)    (1,1,1)
       (1,2)  (1,1,1)  (1,2,3)  (1,1,2)    (1,1,2)
       (2,1)  (2,2,1)  (1,3,2)  (1,2,2)    (1,2,2)
              (3,2,1)  (2,1,3)  (1,2,3)    (1,2,3)
                       (2,3,1)  (1,3,2)    (1,3,2)
                       (3,2,1)  (2,1,3)    (2,1,1)
                                (2,3,1)    (2,1,2)
                                (3,1,2)    (2,1,3)
                                (3,2,1)    (2,2,1)
                                (2,2,1,1)  (2,3,1)
                                           (3,1,2)
                                           (3,2,1)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Patterns matched by standard compositions are counted by A335454.
Patterns matched by compositions of n are counted by A335456(n).
The version for Heinz numbers of partitions is A335550.
Patterns are counted by A000670 and ranked by A333217.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A334299(n) distinct subsequences.
Cf. A056986, A124767, A124770, A124771, A269134, A333218, A333257, A335549.

A335454 Number of normal patterns matched by the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 5, 3, 6, 5, 5, 2, 3, 3, 5, 3, 5, 6, 7, 3, 6, 5, 9, 5, 9, 7, 6, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 4, 7, 5, 10, 9, 9, 3, 6, 5, 9, 4, 9, 10, 12, 5, 9, 7, 13, 7, 12, 9, 7, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 5, 7, 6, 10, 9, 9, 3, 5, 6, 8, 5

Offset: 0

Gus Wiseman, Jun 14 2020

We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

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 a(n) patterns for n = 0, 1, 3, 7, 11, 13, 23, 83, 27, 45:
  0:  1:   11:   111:   211:   121:   2111:   2311:   1211:   2121:
---------------------------------------------------------------------
  ()  ()   ()    ()     ()     ()     ()      ()      ()      ()
      (1)  (1)   (1)    (1)    (1)    (1)     (1)     (1)     (1)
           (11)  (11)   (11)   (11)   (11)    (11)    (11)    (11)
                 (111)  (21)   (12)   (21)    (12)    (12)    (12)
                        (211)  (21)   (111)   (21)    (21)    (21)
                               (121)  (211)   (211)   (111)   (121)
                                      (2111)  (231)   (121)   (211)
                                              (2311)  (211)   (212)
                                                      (1211)  (221)
                                                              (2121)

John Tyler Rascoe, Table of n, a(n) for n = 0..8192
Wikipedia, Permutation pattern
Gus Wiseman, Statistics, classes, and transformations of standard compositions
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

References found in the links are not all included here.
Summing over indices with binary length n gives A335456(n).
The contiguous case is A335458.
The version for Heinz numbers of partitions is A335549.
Patterns are counted by A000670 and ranked by A333217.
The n-th composition has A124771(n) distinct consecutive subsequences.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A333257(n) distinct subsequence-sums.
The n-th composition has A334299(n) distinct subsequences.
Minimal avoided patterns are counted by A335465.
Cf. A034691, A056986, A108917, A124767, A124770, A158005, A269134, A333218, A333222, A333224, A334030.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Length[Union[mstype/@Subsets[stc[n]]]],{n,0,30}]

from itertools import combinations
def comp(n):
    # see A357625
    return
def A335465(n):
    A,B,C = set(),set(),comp(n)
    c = range(len(C))
    for j in c:
        for k in combinations(c, j):
            A.add(tuple(C[i] for i in k))
    for i in A:
        D = {v: rank + 1 for rank, v in enumerate(sorted(set(i)))}
        B.add(tuple(D[v] for v in i))
    return len(B)+1 # John Tyler Rascoe, Mar 12 2025

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.

A335458 Number of normal patterns contiguously matched by the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 5, 3, 5, 5, 5, 2, 3, 3, 5, 3, 5, 5, 7, 3, 5, 5, 8, 5, 8, 7, 6, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 4, 7, 5, 7, 8, 9, 3, 5, 5, 8, 4, 8, 7, 11, 5, 8, 7, 11, 7, 11, 9, 7, 2, 3, 3, 5, 3, 4, 5, 7, 3, 5, 5, 7, 5, 7, 8, 9, 3, 5, 5, 8, 5, 7

Offset: 0

Gus Wiseman, Jun 21 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.

We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(180) = 7 patterns are: (), (1), (1,2), (2,1), (1,2,3), (2,1,2), (2,1,2,3).

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The non-contiguous version is A335454.
Summing over indices with binary length n gives A335457(n).
The nonempty version is A335474.
Patterns are counted by A000670 and ranked by A333217.
The n-th composition has A124771(n) distinct consecutive subsequences.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A333257(n) distinct subsequence-sums.
The n-th composition has A334299(n) distinct subsequences.
Minimal avoided patterns are counted by A335465.
Cf. A108917, A124767, A124770, A181796, A269134, A333218, A333222, A333628, A334030, A335456.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Length[Union[mstype/@ReplaceList[stc[n],{_,s___,_}:>{s}]]],{n,0,30}]

a(n) = A335474(n) + 1.

A335457 Number of normal patterns contiguously matched by compositions of n.

Original entry on oeis.org

1, 2, 5, 12, 31, 80, 196, 486, 1171, 2787, 6564, 15323, 35403, 81251, 185087, 418918, 942525, 2109143, 4695648, 10405694, 22959156

Offset: 0

Gus Wiseman, Jun 23 2020

We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(0) = 1 through a(3) = 12 pairs of a composition with a contiguously matched pattern:
  ()()  (1)()   (2)()     (3)()
        (1)(1)  (11)()    (12)()
                (2)(1)    (21)()
                (11)(1)   (3)(1)
                (11)(11)  (111)()
                          (12)(1)
                          (21)(1)
                          (111)(1)
                          (12)(12)
                          (21)(21)
                          (111)(11)
                          (111)(111)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for standard compositions is A335458.
The non-contiguous version is A335456.
Patterns are counted by A000670 and ranked by A333217.
The n-th standard composition has A124771(n) contiguous subsequences.
Patterns contiguously matched by prime indices are A335549.
Minimal avoided patterns of prime indices are counted by A335550.
Cf. A000005, A056986, A108917, A124767, A181796, A269134, A333224, A334299.

mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Sum[Length[Union[mstype/@ReplaceList[cmp,{_,s___,_}:>{s}]]],{cmp,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}]

a(16)-a(20) from Jinyuan Wang, Jul 08 2020

A334299 Number of distinct subsequences (not necessarily contiguous) of compositions in standard order (A066099).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 4, 4, 2, 4, 3, 6, 4, 7, 6, 5, 2, 4, 4, 6, 4, 6, 7, 8, 4, 7, 6, 10, 6, 10, 8, 6, 2, 4, 4, 6, 3, 8, 8, 8, 4, 8, 4, 9, 8, 12, 11, 10, 4, 7, 8, 10, 8, 11, 12, 13, 6, 10, 9, 14, 8, 13, 10, 7, 2, 4, 4, 6, 4, 8, 8, 8, 4, 6, 6, 12, 7, 14, 12, 10, 4

Offset: 0

Gus Wiseman, Jun 01 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.

			Triangle begins:
  1
  2
  2 3
  2 4 4 4
  2 4 3 6 4 7 6 5
  2 4 4 6 4 6 7 8 4 7 6 10 6 10 8 6
If the k-th composition in standard order is c, then we say that the STC-number of c is k. The n-th column below lists the STC-numbers of the subsequences of the composition with STC-number n:
  0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
     0  0  1  0  2  2  3  0  4   2   5   4   6   6   7
           0     1  1  1     1   0   3   1   5   3   3
                 0  0  0     0       2   0   3   2   1
                                     1       2   1   0
                                     0       1   0
                                             0

Row lengths are A011782.
Looking only at contiguous subsequences gives A124771.
Compositions where every subinterval has a different sum are A333222.
Knapsack compositions are A333223.
Contiguous positive subsequence-sums are counted by A333224.
Contiguous subsequence-sums are counted by A333257.
Disallowing empty subsequences gives A334300.
Subsequence-sums are counted by A334968.
Cf. A000120, A029931, A048793, A066099, A070939, A108917, A325676, A334967.

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

a(n) = A334300(n) + 1.

A333223 Numbers k such that every distinct consecutive subsequence of the k-th composition in standard order has a different sum.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 20, 21, 24, 26, 28, 31, 32, 33, 34, 35, 36, 40, 41, 42, 48, 50, 56, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 80, 81, 84, 85, 88, 96, 98, 100, 104, 106, 112, 120, 127, 128, 129, 130, 131, 132, 133

Offset: 1

Gus Wiseman, Mar 17 2020

nonn

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 list of terms together with the corresponding compositions begins:
    0: ()            21: (2,2,1)           65: (6,1)
    1: (1)           24: (1,4)             66: (5,2)
    2: (2)           26: (1,2,2)           67: (5,1,1)
    3: (1,1)         28: (1,1,3)           68: (4,3)
    4: (3)           31: (1,1,1,1,1)       69: (4,2,1)
    5: (2,1)         32: (6)               70: (4,1,2)
    6: (1,2)         33: (5,1)             71: (4,1,1,1)
    7: (1,1,1)       34: (4,2)             72: (3,4)
    8: (4)           35: (4,1,1)           73: (3,3,1)
    9: (3,1)         36: (3,3)             74: (3,2,2)
   10: (2,2)         40: (2,4)             80: (2,5)
   12: (1,3)         41: (2,3,1)           81: (2,4,1)
   15: (1,1,1,1)     42: (2,2,2)           84: (2,2,3)
   16: (5)           48: (1,5)             85: (2,2,2,1)
   17: (4,1)         50: (1,3,2)           88: (2,1,4)
   18: (3,2)         56: (1,1,4)           96: (1,6)
   19: (3,1,1)       63: (1,1,1,1,1,1)     98: (1,4,2)
   20: (2,3)         64: (7)              100: (1,3,3)

Distinct subsequences are counted by A124770 and A124771.
A superset of A333222, counted by A169942, with partition case A325768.
These compositions are counted by A325676.
A version for partitions is A325769, with Heinz numbers A325778.
The number of distinct positive subsequence-sums is A333224.
The number of distinct subsequence-sums is A333257.
Numbers whose binary indices are a strict knapsack partition are A059519.
Knapsack partitions are counted by A108917, with strict case A275972.
Golomb subsets are counted by A143823.
Heinz numbers of knapsack partitions are A299702.
Maximal Golomb rulers are counted by A325683.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295 A325779, A233564, A325680, A325687, A325770, A333217.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],UnsameQ@@Total/@Union[ReplaceList[stc[#],{_,s__,_}:>{s}]]&]

cp's OEIS Frontend

A335279 Positions of first appearances in A124771 = number of distinct contiguous subsequences of compositions in standard order.

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A233564 c-squarefree numbers: positive integers which in binary are concatenation of distinct parts of the form 10...0 with nonnegative number of zeros.

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A335456 Number of normal patterns matched by compositions of n.

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A335465 Number of minimal normal patterns avoided by the n-th composition in standard order (A066099).

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A335454 Number of normal patterns matched by the n-th composition in standard order (A066099).

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

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A335458 Number of normal patterns contiguously matched by the n-th composition in standard order (A066099).

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A335457 Number of normal patterns contiguously matched by compositions of n.

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