A333223 -id:A333223 - cp's OEIS frontend

A124770 Number of distinct nonempty subsequences for compositions in standard order.

0, 1, 1, 2, 1, 3, 3, 3, 1, 3, 2, 5, 3, 5, 5, 4, 1, 3, 3, 5, 3, 5, 5, 7, 3, 5, 5, 8, 5, 8, 7, 5, 1, 3, 3, 5, 2, 6, 6, 7, 3, 6, 3, 8, 6, 7, 8, 9, 3, 5, 6, 8, 6, 8, 7, 11, 5, 8, 8, 11, 7, 11, 9, 6, 1, 3, 3, 5, 3, 6, 6, 7, 3, 5, 5, 9, 5, 9, 9, 9, 3, 6, 5, 9, 5, 7, 8, 11, 6, 9, 8, 11, 9, 11, 11, 11, 3, 5, 6, 8, 5, 9

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

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. - Gus Wiseman, Apr 03 2020

			Composition number 11 is 2,1,1; the nonempty subsequences are 1; 2; 1,1; 2,1; 2,1,1; so a(11) = 5.
The table starts:
  0
  1
  1 2
  1 3 3 3
  1 3 2 5 3 5 5 4
  1 3 3 5 3 5 5 7 3 5 5 8 5 8 7 5
From _Gus Wiseman_, Apr 03 2020: (Start)
If the k-th composition in standard order is c, then we say that the STC-number of c is k. The STC-numbers of the distinct subsequences of the composition with STC-number k are given in column k below:
  1  2  1  4  1  1  1  8  1  2   1   1   1   1   1   16  1   2   1   2
        3     2  2  3     4  10  2   4   2   2   3       8   4   4   4
              5  6  7     9      3   12  6   3   7       17  18  3   20
                                 5       5   6   15              9
                                 11      13  14                  19
(End)

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

Row lengths are A011782.
Allowing empty subsequences gives A124771.
Dominates A333224, the version counting subsequence-sums instead of subsequences.
Compositions where every restriction to a subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222. The case of partitions is counted by A325768 and ranked by A325779.
Positive subset-sums of partitions are counted by A276024 and A299701.
Knapsack compositions are counted by A325676 and A325687 and ranked by A333223. The case of partitions is counted by A325769 and ranked by A325778, for which the number of distinct consecutive subsequences is given by A325770.
Cf. A000120, A003022, A029931, A048793, A066099, A070939, A103295, A108917, A143823, A325680.

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, Apr 03 2020 *)

a(n) = A124771(n) - 1. - Gus Wiseman, Apr 03 2020

A353390 Number of compositions of n whose own run-lengths are a subsequence (not necessarily consecutive).

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 3, 2, 2, 8, 17, 26, 43, 77, 129, 210, 351, 569

Offset: 0

Gus Wiseman, May 15 2022

			The a(0) = 1 through a(9) = 8 compositions (empty columns indicated by dots):
  ()  (1)  .  .  (22)  (122)  (1122)  (11221)  (21122)  (333)
                       (221)  (1221)  (12211)  (22112)  (22113)
                              (2211)                    (22122)
                                                        (31122)
                                                        (121122)
                                                        (122112)
                                                        (211221)
                                                        (221121)
For example, the composition y = (2,2,3,3,1) has run-lengths (2,2,1), which form a (non-consecutive) subsequence, so y is counted under a(11).

The version for partitions is A325702.
The recursive version is A353391, ranked by A353431.
The consecutive case is A353392, ranked by A353432.
These compositions are ranked by A353402.
The reverse version is A353403.
The recursive consecutive version is A353430.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A047966 counts uniform partitions, compositions A329738.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223, partitions A108917.
A325705 counts partitions containing all of their distinct multiplicities.
A329739 counts compositions with all distinct run-lengths, for runs A351013.
A353400 counts compositions with all run-lengths > 2.
Cf. A005811, A103295, A114901, A181591, A238279, A242882, A324572, A333755, A351017, A353401, A353426.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MemberQ[Subsets[#],Length/@Split[#]]&]],{n,0,15}]

A353391 Number of compositions of n that are empty, a singleton, or whose run-lengths are a subsequence that is already counted.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 5, 7, 9, 11, 15, 22, 38, 45, 87, 93

Offset: 0

Gus Wiseman, May 15 2022

			The a(9) = 4 through a(14) = 15 compositions (A..E = 10..14):
  (9)       (A)       (B)       (C)       (D)       (E)
  (333)     (2233)    (141122)  (2244)    (161122)  (2255)
  (121122)  (3322)    (221123)  (4422)    (221125)  (5522)
  (221121)  (131122)  (221132)  (151122)  (221134)  (171122)
            (221131)  (221141)  (221124)  (221143)  (221126)
                      (231122)  (221142)  (221152)  (221135)
                      (321122)  (221151)  (221161)  (221153)
                                (241122)  (251122)  (221162)
                                (421122)  (341122)  (221171)
                                          (431122)  (261122)
                                          (521122)  (351122)
                                                    (531122)
                                                    (621122)
                                                    (122121122)
                                                    (221121221)

The non-recursive version is A353390, ranked by A353402.
The non-recursive consecutive version is A353392, ranked by A353432.
The non-recursive reverse version is A353403.
The unordered version is A353426, ranked by A353393 (nonprime A353389).
The consecutive version is A353430.
These compositions are ranked by A353431.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A329738 counts uniform compositions, partitions A047966.
A114901 counts compositions with no runs of length 1.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223.
A325705 counts partitions containing all of their distinct multiplicities.
A329739 counts compositions with all distinct run-length.
Cf. A005811, A032020, A103295, A114640, A165413, A181591, A242882, A324572, A325702, A333755, A351013, A353401.

yosQ[y_]:=Length[y]<=1||MemberQ[Subsets[y],Length/@Split[y]]&&yosQ[Length/@Split[y]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],yosQ]],{n,0,15}]

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[#]]&]

A353392 Number of compositions of n whose own run-lengths are a consecutive subsequence.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 2, 2, 2, 8, 12, 16, 20, 35, 46, 59, 81, 109, 144, 202, 282

Offset: 0

Gus Wiseman, May 15 2022

			The a(0) = 0 through a(10) = 12 compositions (empty columns indicated by dots, 0 is the empty composition):
  0  1  .  .  22  122  1122  11221  21122  333     1333
                  221  2211  12211  22112  22113   2233
                                           22122   3322
                                           31122   3331
                                           121122  22114
                                           122112  41122
                                           211221  122113
                                           221121  131122
                                                   221131
                                                   311221
                                                   1211221
                                                   1221121

The non-consecutive version for partitions is A325702.
The non-consecutive version is A353390, ranked by A353402.
The non-consecutive recursive version is A353391, ranked by A353431.
The non-consecutive reverse version is A353403.
The recursive version is A353430.
These compositions are ranked by A353432.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223.
A329738 counts uniform compositions, partitions A047966.
A329739 counts compositions with all distinct run-lengths.
Cf. A008965, A032020, A103295, A103300, A114901, A238279, A324572, A325705, A333224, A333755, A351013, A353401.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],#=={}||MemberQ[Join@@Table[Take[#,{i,j}],{i,Length[#]},{j,i,Length[#]}],Length/@Split[#]]&]],{n,0,15}]

A353403 Number of compositions of n whose own reversed run-lengths are a subsequence (not necessarily consecutive).

Original entry on oeis.org

1, 1, 0, 0, 3, 2, 5, 12, 16, 30, 45, 94, 159, 285, 477, 864, 1487, 2643

Offset: 0

Gus Wiseman, May 15 2022

			The a(0) = 1 through a(7) = 12 compositions:
  ()  (1)  .  .  (22)   (1121)  (1113)  (1123)
                 (112)  (1211)  (1122)  (1132)
                 (211)          (1221)  (2311)
                                (2211)  (3211)
                                (3111)  (11131)
                                        (11212)
                                        (11221)
                                        (12112)
                                        (12211)
                                        (13111)
                                        (21121)
                                        (21211)

The non-reversed version is A353390, ranked by A353402, partitions A325702.
The non-reversed recursive version is A353391, ranked by A353431.
The non-reversed consecutive case is A353392, ranked by A353432.
The non-reversed recursive consecutive version is A353430.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223, partitions A108917.
A325705 counts partitions containing all of their distinct multiplicities.
A329739 counts compositions with all distinct run-lengths, for runs A351013.
Cf. A005811, A032020, A103295, A114640, A165413, A324572, A333755, A353400, A353401, A353426.

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],MemberQ[Subsets[#],Reverse[Length/@Split[#]]]&]],{n,0,15}]

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[#]]&]

A353430 Number of integer compositions of n that are empty, a singleton, or whose own run-lengths are a consecutive subsequence that is already counted.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 5, 7, 9, 11, 15, 16, 22, 25, 37, 37, 45

Offset: 0

Gus Wiseman, May 16 2022

			The a(n) compositions for selected n (A..E = 10..14):
  n=4:  n=6:    n=9:      n=10:     n=12:     n=14:
-----------------------------------------------------------
  (4)   (6)     (9)       (A)       (C)       (E)
  (22)  (1122)  (333)     (2233)    (2244)    (2255)
        (2211)  (121122)  (3322)    (4422)    (5522)
                (221121)  (131122)  (151122)  (171122)
                          (221131)  (221124)  (221126)
                                    (221142)  (221135)
                                    (221151)  (221153)
                                    (241122)  (221162)
                                    (421122)  (221171)
                                              (261122)
                                              (351122)
                                              (531122)
                                              (621122)
                                              (122121122)
                                              (221121221)

Non-recursive non-consecutive version: counted by A353390, ranked by A353402, reverse A353403, partitions A325702.
Non-consecutive version: A353391, ranked by A353431, partitions A353426.
Non-recursive version: A353392, ranked by A353432.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A114901 counts compositions with no runs of length 1.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223.
A329738 counts uniform compositions, partitions A047966.
A329739 counts compositions with all distinct run-lengths.
Cf. A005811, A032020, A103295, A114640, A165413, A242882, A325705, A333755, A351013, A353400, A353401.

yoyQ[y_]:=Length[y]<=1||MemberQ[Join@@Table[Take[y,{i,j}],{i,Length[y]},{j,i,Length[y]}],Length/@Split[y]]&&yoyQ[Length/@Split[y]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],yoyQ]],{n,0,15}]

A354580 Number of rucksack compositions of n: every distinct partial run has a different sum.

Original entry on oeis.org

1, 1, 2, 4, 6, 12, 22, 39, 68, 125, 227, 402, 710, 1280, 2281, 4040, 7196, 12780, 22623, 40136, 71121, 125863, 222616, 393305, 695059, 1227990, 2167059, 3823029, 6743268, 11889431, 20955548, 36920415, 65030404, 114519168, 201612634, 354849227

Offset: 0

Gus Wiseman, Jun 13 2022

nonn

We define a partial run of a sequence to be any contiguous constant subsequence. The term rucksack is short for run-knapsack.

			The a(0) = 1 through a(5) = 12 compositions:
  ()  (1)  (2)    (3)      (4)        (5)
           (1,1)  (1,2)    (1,3)      (1,4)
                  (2,1)    (2,2)      (2,3)
                  (1,1,1)  (3,1)      (3,2)
                           (1,2,1)    (4,1)
                           (1,1,1,1)  (1,1,3)
                                      (1,2,2)
                                      (1,3,1)
                                      (2,1,2)
                                      (2,2,1)
                                      (3,1,1)
                                      (1,1,1,1,1)

Max Alekseyev, Table of n, a(n) for n = 0..65

The knapsack version is A325676, ranked by A333223.
The non-partial version for partitions is A353837, ranked by A353838 (complement A353839).
The non-partial version is A353850, ranked by A353852.
The version for partitions is A353864, ranked by A353866.
The complete version for partitions is A353865, ranked by A353867.
These compositions are ranked by A354581.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A108917 counts knapsack partitions, ranked by A299702, strict A275972.
A238279 and A333755 count compositions by number of runs.
A275870 counts collapsible partitions, ranked by A300273.
A353836 counts partitions by number of distinct run-sums.
A353847 is the composition run-sum transformation.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions, ranked by A354908.
Cf. A143823, A169942, A242882, A325545, A325680, A325682, A325685, A325687, A329739, A351017, A353849.

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],UnsameQ@@Total/@Union@@Subsets/@Split[#]&]],{n,0,15}]

Terms a(16) onward from Max Alekseyev, Sep 10 2023

A334967 Numbers k such that the every subsequence (not necessarily contiguous) of the k-th composition in standard order (A066099) 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, 42, 48, 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, 134

Offset: 1

Gus Wiseman, Jun 02 2020

nonn

First differs from A333223 in lacking 41.

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: ()           18: (3,2)          48: (1,5)
   1: (1)          19: (3,1,1)        56: (1,1,4)
   2: (2)          20: (2,3)          63: (1,1,1,1,1,1)
   3: (1,1)        21: (2,2,1)        64: (7)
   4: (3)          24: (1,4)          65: (6,1)
   5: (2,1)        26: (1,2,2)        66: (5,2)
   6: (1,2)        28: (1,1,3)        67: (5,1,1)
   7: (1,1,1)      31: (1,1,1,1,1)    68: (4,3)
   8: (4)          32: (6)            69: (4,2,1)
   9: (3,1)        33: (5,1)          70: (4,1,2)
  10: (2,2)        34: (4,2)          71: (4,1,1,1)
  12: (1,3)        35: (4,1,1)        72: (3,4)
  15: (1,1,1,1)    36: (3,3)          73: (3,3,1)
  16: (5)          40: (2,4)          74: (3,2,2)
  17: (4,1)        42: (2,2,2)        80: (2,5)

These compositions are counted by A334268.
Golomb rulers are counted by A169942 and ranked by A333222.
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 ranked by A333223.
The case of partitions is counted by A325769 and ranked by A325778.
Contiguous subsequence-sums are counted by A333224 and ranked by A333257.
Number of (not necessarily contiguous) subsequences is A334299.
Cf. A000120, A029931, A048793, A066099, A070939, A108917, A124771, A325770, A334300, A334967.

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

cp's OEIS Frontend

A124770 Number of distinct nonempty subsequences for compositions in standard order.

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A353390 Number of compositions of n whose own run-lengths are a subsequence (not necessarily consecutive).

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A353391 Number of compositions of n that are empty, a singleton, or whose run-lengths are a subsequence that is already counted.

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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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A353392 Number of compositions of n whose own run-lengths are a consecutive subsequence.

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A353403 Number of compositions of n whose own reversed run-lengths are a subsequence (not necessarily consecutive).

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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.

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A353430 Number of integer compositions of n that are empty, a singleton, or whose own run-lengths are a consecutive subsequence that is already counted.

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A354580 Number of rucksack compositions of n: every distinct partial run has a different sum.

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