A353932 -id:A353932 - cp's OEIS frontend

A353856 Triangle read by rows where T(n,k) is the number of integer compositions of n with run-sum trajectory (condensation) ending in a composition of length k.

1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 5, 2, 1, 0, 0, 2, 12, 2, 0, 0, 0, 8, 10, 12, 2, 0, 0, 0, 2, 32, 23, 6, 1, 0, 0, 0, 20, 26, 51, 28, 3, 0, 0, 0, 0, 5, 66, 109, 52, 22, 2, 0, 0, 0, 0, 8, 108, 144, 188, 53, 10, 1, 0, 0, 0, 0, 2, 134, 358, 282, 196, 48, 4, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Jun 01 2022

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 run-sum trajectory is obtained by repeatedly taking the run-sums transformation (or condensation, represented by A353847) until an anti-run is reached. For example, the trajectory (2,1,1,3,1,1,2,1,1,2,1) -> (2,2,3,2,2,2,2,1) -> (4,3,8,1) is counted under T(15,4).

			Triangle begins:
   1
   0   1
   0   2   0
   0   2   2   0
   0   5   2   1   0
   0   2  12   2   0   0
   0   8  10  12   2   0   0
   0   2  32  23   6   1   0   0
   0  20  26  51  28   3   0   0   0
   0   5  66 109  52  22   2   0   0   0
   0   8 108 144 188  53  10   1   0   0   0
   0   2 134 358 282 196  48   4   0   0   0   0
For example, row n = 6 counts the following compositions:
  .  (6)       (15)     (123)    (1212)  .  .
     (33)      (24)     (132)    (2121)
     (222)     (42)     (141)
     (1113)    (51)     (213)
     (2112)    (114)    (231)
     (3111)    (411)    (312)
     (11211)   (1122)   (321)
     (111111)  (2211)   (1131)
               (11112)  (1221)
               (21111)  (1311)
                        (11121)
                        (12111)

Row sums are A011782.
Row-lengths without zeros appear to be A131737.
The version for partitions is A353843.
The length of the trajectory is A353854, firsts A072639, partitions A353841.
The last part of the same trajectory is A353855.
Column k = 1 is A353858.
A066099 lists compositions in standard order.
A318928 gives runs-resistance of binary expansion.
A325268 counts partitions by omicron, rank statistic A304465.
A333489 ranks anti-runs, counted by A003242 (complement A261983).
A333627 ranks the run-lengths of standard compositions.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sums of a composition, partitions A353832.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions.
Cf. A237685, A238279, A304442, A325277, A333755, A353848, A353850, A353852.

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],Length[FixedPoint[Total/@Split[#]&,#]]==k&]],{n,0,15},{k,0,n}]

A354581 Numbers k such that the k-th composition in standard order is rucksack, meaning every distinct partial run has a different sum.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 44, 45, 48, 49, 50, 51, 52, 53, 54, 56, 57, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 80, 81, 82, 84, 85, 86, 88

Offset: 0

Gus Wiseman, Jun 15 2022

nonn

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

			The terms together with their corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   5: (2,1)
   6: (1,2)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  10: (2,2)
  12: (1,3)
  13: (1,2,1)
  15: (1,1,1,1)
Missing are:
  11: (2,1,1)
  14: (1,1,2)
  23: (2,1,1,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  30: (1,1,1,2)
  39: (3,1,1,1)
  43: (2,2,1,1)
  46: (2,1,1,2)

The version for binary indices is A000225.
Counting distinct sums of full runs gives A353849, partitions A353835.
For partitions we have A353866, counted by A353864, complement A354583.
These compositions are counted by A354580.
Counting distinct sums of partial runs gives A354907, partitions A353861.
A066099 lists all compositions in standard order.
A124767 counts runs in standard compositions.
A124771 counts distinct contiguous subsequences, non-contiguous A334299.
A238279 and A333755 count compositions by number of runs.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions, rows ranked by A353847.
Cf. A000120, A005811, A029837, A063787, A175413, A181819, A330036, A333381, A333489, A353832, A353860.

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

A357850 Numbers whose prime indices do not have weakly decreasing run-sums. Heinz numbers of the partitions counted by A357865.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 26, 28, 30, 33, 34, 35, 36, 38, 39, 42, 44, 46, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 98, 99, 100, 102, 104, 105, 106, 108, 110, 111

Offset: 1

Gus Wiseman, Oct 19 2022

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. 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 terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}
   33: {2,5}
   34: {1,7}
   35: {3,4}
   36: {1,1,2,2}
   38: {1,8}
   39: {2,6}
   42: {1,2,4}

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

These are the indices of rows in A354584 that are not weakly decreasing.
The complement is A357861, counted by A304406.
These partitions are counted by A357865.
The opposite (not weakly increasing) version is A357876, counted by A357878.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A118914, A181819, A300273, A304405, A304428, A304430, A304442, A353832, A353864, A353932, A357864, A357875.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!GreaterEqual@@Total/@Split[primeMS[#]]&]

A353843 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory ending in a partition of length k. All zeros removed.

Original entry on oeis.org

1, 1, 2, 2, 1, 4, 1, 2, 5, 5, 5, 1, 2, 12, 1, 8, 11, 3, 3, 19, 8, 5, 27, 9, 1, 2, 34, 19, 1, 15, 26, 34, 2, 2, 49, 45, 5, 5, 68, 48, 14, 4, 58, 98, 15, 1, 18, 76, 105, 31, 1, 2, 88, 159, 46, 2, 13, 98, 191, 79, 4, 2, 114, 261, 105, 8, 14, 148, 282, 164, 19

Offset: 0

Gus Wiseman, Jun 04 2022

The partition run-sum trajectory is obtained by repeatedly taking the run-sums until a strict partition is reached. For example, the trajectory of y = (3,2,1,1,1) is (3,2,1,1,1) -> (3,3,2) -> (6,2), so y is counted under T(8,2).

			Triangle begins:
   1
   1
   2
   2  1
   4  1
   2  5
   5  5  1
   2 12  1
   8 11  3
   3 19  8
   5 27  9  1
   2 34 19  1
  15 26 34  2
   2 49 45  5
   5 68 48 14
   4 58 98 15  1
For example, row n = 8 counts the following partitions:
  (8)         (53)       (431)
  (44)        (62)       (521)
  (422)       (71)       (3221)
  (2222)      (332)
  (4211)      (611)
  (41111)     (3311)
  (221111)    (5111)
  (11111111)  (22211)
              (32111)
              (311111)
              (2111111)

Row sums are A000041.
Row-lengths are A003056.
The last part of the same trajectory is A353842.
Column k = 1 is A353845, compositions A353858.
The length of the trajectory is A353846.
The version for compositions is A353856.
A275870 counts collapsible partitions, ranked by A300273.
A304442 counts partitions with constant run-sums, ranked by A353833/A353834.
A325268 counts partitions by omicron, rank statistic A304465.
A353837 counts partitions with all distinct run-sums, ranked by A353838.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sums of a composition, partitions A353832.
A353864 counts rucksack partitions, ranked by A353866.
A353865 counts perfect rucksack partitions, ranked by A353867.
A353932 lists run-sums of standard compositions.
Cf. A008284, A116608, A325242, A325268, A225485 or A325280.
Cf. A047966, A237685, A325277, A353841, A353853-A353859.

Table[Length[Select[IntegerPartitions[n], Length[FixedPoint[Sort[Total/@Split[#]]&,#]]==k&]],{n,0,15},{k,0,n}]

A353857 Numbers k such that the k-th composition in standard order has run-sum trajectory ending in a singleton.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 31, 32, 36, 39, 42, 46, 59, 60, 63, 64, 127, 128, 136, 138, 139, 142, 143, 168, 170, 174, 175, 184, 186, 187, 232, 238, 239, 248, 250, 251, 255, 256, 292, 316, 487, 511, 512, 528, 543, 682, 750, 955, 1008, 1023, 1024, 2047

Offset: 1

Gus Wiseman, Jun 01 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 run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353847) until the rank of an anti-run is reached. For example, the trajectory 11 -> 10 -> 8 corresponds to the trajectory (2,1,1) -> (2,2) -> (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.

			The terms together with their binary expansions and corresponding compositions begin:
   1:        1  (1)
   2:       10  (2)
   3:       11  (1,1)
   4:      100  (3)
   7:      111  (1,1,1)
   8:     1000  (4)
  10:     1010  (2,2)
  11:     1011  (2,1,1)
  14:     1110  (1,1,2)
  15:     1111  (1,1,1,1)
  16:    10000  (5)
  31:    11111  (1,1,1,1,1)
  32:   100000  (6)
  36:   100100  (3,3)
  39:   100111  (3,1,1,1)
  42:   101010  (2,2,2)
  46:   101110  (2,1,1,2)
  59:   111011  (1,1,2,1,1)
  60:   111100  (1,1,1,3)
  63:   111111  (1,1,1,1,1,1)

The version for partitions is A353844.
The trajectory length is A353854, firsts A072639, for partitions A353841.
The last part of the trajectory is A353855, for partitions A353842.
These compositions are counted by A353858.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order.
A318928 gives runs-resistance of binary expansion.
A325268 counts partitions by omicron, rank statistic A304465.
A333627 ranks the run-lengths of standard compositions.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents composition run-sum transformation, partitions A353832.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions.
Cf. A237685, A238279, A304442, A325277, A333381, A333755, A353848, A353849, A353850, A353852.

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

A354582 Number of distinct contiguous constant subsequences (or partial runs) in the k-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 13 2022

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 number 981 in standard order is (1,1,1,2,2,2,1), with partial runs (1), (2), (1,1), (2,2), (1,1,1), (2,2,2), so a(981) = 6.
As a triangle:
  1
  1 2
  1 2 2 3
  1 2 2 3 2 2 3 4
  1 2 2 3 2 3 2 4 2 2 3 3 3 3 4 5
  1 2 2 3 2 3 3 4 2 3 3 4 3 2 3 5 2 2 3 3 3 3 2 4 3 3 4 3 4 4 5 6

The version for partitions is A001222, full A001221.
If we allow any constant subsequence we get A063787.
If we allow any contiguous subsequence we get A124771.
Positions of first appearances are A126646.
The version for binary indices is A330036, full A005811.
If we allow any subsequence we get A334299.
The full version is A351014, firsts A351015.
The version for run-sums of partitions is A353861, full A353835.
Counting distinct sums of partial runs gives A354907, full A353849.
A066099 lists all compositions in standard order.
A124767 counts runs in standard compositions.
A238279 and A333755 count compositions by number of runs.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions, rows ranked by A353847.
Cf. A000120, A003242, A029837, A175413, A274174, A333381, A333489, A353832, A353860, A353864.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pre[y_]:=NestWhileList[Most,y,Length[#]>1&];
Table[Length[Union[Join@@pre/@Split[stc[n]]]],{n,0,100}]

A354909 Number of integer compositions of n that are not the run-sums of any other composition.

Original entry on oeis.org

0, 0, 1, 1, 3, 7, 16, 33, 74, 155, 329, 688, 1439, 2975, 6154, 12654, 25964, 53091, 108369, 220643, 448520

Offset: 0

Gus Wiseman, Jun 19 2022

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 a(0) = 0 through a(6) = 16 compositions:
  .  .  (11)  (111)  (112)   (113)    (114)
                     (211)   (311)    (411)
                     (1111)  (1112)   (1113)
                             (1121)   (1122)
                             (1211)   (1131)
                             (2111)   (1221)
                             (11111)  (1311)
                                      (2112)
                                      (2211)
                                      (3111)
                                      (11112)
                                      (11121)
                                      (11211)
                                      (12111)
                                      (21111)
                                      (111111)

The version for binary words is A000918, complement A000126.
These compositions are ranked by A354904 = positions of zeros in A354578.
The complement is counted by A354910, ranked by A354912.
A003242 counts anti-run compositions, ranked by A333489.
A238279 and A333755 count compositions by number of runs.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions, rows ranked by A353847.
Cf. A005811, A027336, A066099, A124767, A274174, A351597, A353849, A353850, A353860, A354905, A354907.

Table[Length[Complement[Join@@Permutations/@IntegerPartitions[n], Total/@Split[#]&/@Join@@Permutations/@IntegerPartitions[n]]],{n,0,15}]

A354910 Number of compositions of n that are the run-sums of some other composition.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 16, 31, 54, 101, 183, 336, 609, 1121, 2038, 3730, 6804, 12445, 22703, 41501, 75768

Offset: 0

Gus Wiseman, Jun 20 2022

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 a(0) = 0 through a(6) = 16 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)
                (12)  (13)   (14)   (15)
                (21)  (22)   (23)   (24)
                      (31)   (32)   (33)
                      (121)  (41)   (42)
                             (122)  (51)
                             (131)  (123)
                             (212)  (132)
                             (221)  (141)
                                    (213)
                                    (222)
                                    (231)
                                    (312)
                                    (321)
                                    (1212)
                                    (2121)

The version for binary words is A000126, complement A000918
The complement is counted by A354909, ranked by A354904.
These compositions are ranked by A354912 = nonzeros of A354578.
A003242 counts anti-run compositions, ranked by A333489.
A238279 and A333755 count compositions by number of runs.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353853-A353859 pertain to composition run-sum trajectory.
A353932 lists run-sums of standard compositions, rows ranked by A353847.
Cf. A005811, A027336, A066099, A239312, A274174, A351014, A351597, A353849, A353850, A353864, A354905, A354907.

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

A354908 Numbers k such that the k-th composition in standard order (graded reverse-lexicographic, A066099) is collapsible.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 31, 32, 36, 39, 42, 43, 46, 47, 58, 59, 60, 62, 63, 64, 127, 128, 136, 138, 139, 142, 143, 168, 170, 171, 174, 175, 184, 186, 187, 190, 191, 232, 234, 235, 238, 239, 248, 250, 251, 254, 255, 256, 292, 295, 316, 319, 484

Offset: 1

Gus Wiseman, Jun 23 2022

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.

If a collapse is an adding together of some partial run of an integer composition c, we say c is collapsible iff by some sequence of collapses it can be reduced to a single part. An example of such a sequence of collapses is (11132112) -> (332112) -> (33222) -> (6222) -> (66) -> (n), which shows that (11132112) is a collapsible composition of 12.

			The terms together with their corresponding compositions begin:
  1:(1)  2:(2)   4:(3)     8:(4)     16:(5)      32:(6)
         3:(11)  7:(111)  10:(22)    31:(11111)  36:(33)
                          11:(211)               39:(3111)
                          14:(112)               42:(222)
                          15:(1111)              43:(2211)
                                                 46:(2112)
                                                 47:(21111)
                                                 58:(1122)
                                                 59:(11211)
                                                 60:(1113)
                                                 62:(11112)
                                                 63:(111111)

The standard compositions used here are A066099, run-sums A353847/A353932.
The version for Heinz numbers of partitions is A300273, counted by A275870.
These compositions are counted by A353860.
A003242 counts anti-run compositions, ranked by A333489, complement A261983.
A011782 counts compositions.
A124767 counts runs in standard compositions.
A238279 and A333755 count compositions by number of runs.
A334968 counts distinct sums of subsequences of standard compositions.
A351014 counts distinct runs of standard compositions, firsts A351015.
A353853-A353859 pertain to composition run-sum trajectory.
A354582 counts distinct partial runs of standard compositions, sums A354907.
Cf. A005811, A029837, A072639, A124771, A274174, A318928, A333381, A353848, A353849, A353850.

repcams[q_List]:=repcams[q]=Union[{q},If[UnsameQ@@q,{},Union@@repcams/@Union[Insert[Drop[q,#],Plus@@Take[q,#],First[#]]&/@Select[Tuples[Range[Length[q]],2],And[Less@@#,SameQ@@Take[q,#]]&]]]];
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MemberQ[repcams[stc[#]],{_}]&]

A354906 Position of first appearance of n in A354579 = Number of distinct run-lengths of standard compositions.

Original entry on oeis.org

0, 1, 11, 119, 5615, 251871

Offset: 0

Gus Wiseman, Jun 23 2022

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 terms together with their corresponding compositions begin:
       0: ()
       1: (1)
      11: (2,1,1)
     119: (1,1,2,1,1,1)
    5615: (2,2,1,1,1,2,1,1,1,1)
  251871: (1,1,1,2,2,1,1,1,1,2,1,1,1,1,1)

The standard compositions used here are A066099, run-sums A353847/A353932.
The version for partitions is A006939, for run-sums A002110.
For run-sums instead of run-lengths we have A246534 (firsts in A353849).
For runs instead of run-lengths we have A351015 (firsts in A351014).
These are the positions of first appearances in A354579.
A005811 counts runs in binary expansion.
A333627 ranks the run-lengths of standard compositions.
A351596 ranks compositions with distinct run-lengths, counted by A329739.
A353744 ranks compositions with equal run-lengths, counted by A329738.
A353852 ranks compositions with distinct run-sums, counted by A353850.
A353853-A353859 are sequences pertaining to composition run-sum trajectory.
A353860 counts collapsible compositions.
Cf. A003242, A029837, A071625, A124767, A182857, A238279/A333755, A325278, A333381, A333489, A333629.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
pd=Table[Length[Union[Length/@Split[stc[n]]]],{n,0,10000}];
Table[Position[pd,n][[1,1]]-1,{n,0,Max@@pd}]

cp's OEIS Frontend

A353856 Triangle read by rows where T(n,k) is the number of integer compositions of n with run-sum trajectory (condensation) ending in a composition of length k.

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A354581 Numbers k such that the k-th composition in standard order is rucksack, meaning every distinct partial run has a different sum.

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A357850 Numbers whose prime indices do not have weakly decreasing run-sums. Heinz numbers of the partitions counted by A357865.

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A353843 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory ending in a partition of length k. All zeros removed.

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A353857 Numbers k such that the k-th composition in standard order has run-sum trajectory ending in a singleton.

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A354582 Number of distinct contiguous constant subsequences (or partial runs) in the k-th composition in standard order.

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A354909 Number of integer compositions of n that are not the run-sums of any other composition.

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A354910 Number of compositions of n that are the run-sums of some other composition.

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A354908 Numbers k such that the k-th composition in standard order (graded reverse-lexicographic, A066099) is collapsible.

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