A353852 -id:A353852 - cp's OEIS frontend

A354905 First position of n in A354578, where A354578(k) is the number of integer compositions whose run-sums constitute the k-th composition in standard order (graded reverse-lexicographic, A066099).

3, 0, 2, 8, 32, 68, 130, 290, 274, 580, 520, 1298, 2080, 1096, 2082, 4168, 2178, 4164, 4386, 35137, 8328, 8786, 10274, 8772, 16712, 20562, 8712, 16658, 33320, 41554, 33288, 82210, 34856, 66628, 33312, 66642, 34850, 69704, 140306, 133448, 69714, 74308, 133154

Offset: 0

Gus Wiseman, Jun 21 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.

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 terms and their corresponding compositions begin:
     3: (1,1)
     0: ()
     2: (2)
     8: (4)
    32: (6)
    68: (4,3)
   130: (6,2)
   290: (3,4,2)
   274: (4,3,2)
   580: (3,4,3)
   520: (6,4)
  1298: (2,4,3,2)
The inverse run-sum compositions for n = 2, 8, 32, 68, 130, 290:
  (2)   (4)     (6)       (43)     (62)       (342)
  (11)  (22)    (33)      (223)    (332)      (3411)
        (1111)  (222)     (4111)   (611)      (11142)
                (111111)  (11113)  (3311)     (32211)
                          (22111)  (22211)    (111411)
                                   (1111112)  (311112)
                                              (1112211)

The standard compositions used here are A066099, run-sums A353847/A353932.
This is the position of the first appearance of n in A354578.
A011782 counts compositions.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
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.
A354904 lists positions of zeros in A354578, complement A354912.
A354908 ranks collapsible compositions, counted by A353860.
Cf. A000005, A029837, A124767, A238279/A333755, A274174, A333381, A353832, A353849, A353863, A354584.

nn=1000;
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
antirunQ[y_]:=Length[Split[y]]==Length[y];
q=Table[Length[Select[Tuples[Divisors/@stc[n]],antirunQ]],{n,0,nn}];
w=Last[Select[Table[Take[q+1,i],{i,nn}],Union[#]==Range[Max@@#]&]-1];
Table[Position[w,k][[1,1]]-1,{k,0,Max@@w}]

A355748 Number of ways to choose a sequence of divisors, one of each part of the n-th composition in standard order.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 4, 2, 2, 2, 2, 1, 2, 3, 4, 2, 4, 4, 4, 2, 3, 2, 4, 2, 2, 2, 2, 1, 4, 2, 6, 3, 4, 4, 4, 2, 6, 4, 8, 4, 4, 4, 4, 2, 2, 3, 4, 2, 4, 4, 4, 2, 3, 2, 4, 2, 2, 2, 2, 1, 2, 4, 4, 2, 6, 6, 6, 3, 6, 4, 8, 4, 4, 4, 4, 2, 4, 6, 8, 4, 8, 8, 8

Offset: 0

Gus Wiseman, Jul 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.

			Composition number 152 in standard order is (3,1,4), and the a(152) = 6 choices are: (1,1,1), (1,1,2), (1,1,4), (3,1,1), (3,1,2), (3,1,4).

Positions of 1's are A000079 (after the first).
The anti-run case is A354578, zeros A354904, firsts A354905.
An unordered version (using prime indices) is A355731:
- firsts A355732,
- resorted A355733,
- weakly increasing A355735,
- relatively prime A355737,
- strict A355739.
A000005 counts divisors.
A003963 multiplies together the prime indices of n.
A005811 counts runs in binary expansion.
A029837 adds up standard compositions, lengths A000120.
A066099 lists the compositions in standard order.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
Cf. A124767, A175413, A238279, A274174, A326841, A333381, A333755, A353847, A353849, A355747.

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

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.

Original entry on oeis.org

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[#]]&]

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&]

A354579 Number of distinct lengths of runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 11 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 lengths (2,3,1,2).

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 positions of first appearances together with the corresponding compositions begin:
       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)

Standard compositions are listed by A066099.
The version for partitions is A071625.
For runs instead of run-lengths we have A351014, firsts A351015.
Positions of 0's and 1's are A353744, counted by A329738.
For sums instead of lengths we have A353849, ones at A353848.
Positions of first appearances are A354906.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A333627 ranks the run-lengths of standard compositions.
A351596 ranks compositions with distinct run-lengths, counted by A329739.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353847 ranks the run-sums of standard compositions.
A353852 ranks compositions with distinct run-sums, counted by A353850.
A353860 counts collapsible compositions.
Cf. A029837, A124767, A181819, A238279, A333381, A333755, A353839, A353851.

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

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}]

A354583 Heinz numbers of non-rucksack partitions: not every prime-power divisor has a different sum of prime indices.

Original entry on oeis.org

12, 24, 36, 40, 48, 60, 63, 72, 80, 84, 96, 108, 112, 120, 126, 132, 144, 156, 160, 168, 180, 189, 192, 200, 204, 216, 224, 228, 240, 252, 264, 276, 280, 288, 300, 312, 315, 320, 324, 325, 336, 348, 351, 352, 360, 372, 378, 384, 396, 400, 408, 420, 432, 440

Offset: 1

Gus Wiseman, Jun 15 2022

nonn

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 term rucksack is short for run-knapsack.

			The terms together with their prime indices begin:
   12: {1,1,2}
   24: {1,1,1,2}
   36: {1,1,2,2}
   40: {1,1,1,3}
   48: {1,1,1,1,2}
   60: {1,1,2,3}
   63: {2,2,4}
   72: {1,1,1,2,2}
   80: {1,1,1,1,3}
   84: {1,1,2,4}
   96: {1,1,1,1,1,2}
  108: {1,1,2,2,2}
  112: {1,1,1,1,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  144: {1,1,1,1,2,2}
  156: {1,1,2,6}
  160: {1,1,1,1,1,3}
  168: {1,1,1,2,4}
For example, {2,2,2,3,3} does not have distinct run-sums because 2+2+2 = 3+3, so 675 is in the sequence.

Knapsack partitions are counted by A108917, ranked by A299702.
Non-knapsack partitions are ranked by A299729.
The non-partial version is A353839, complement A353838 (counted by A353837).
The complement is A353866, counted by A353864.
The complete complement is A353867, counted by A353865.
The complement for compositions is counted by A354580.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A073093 counts prime-power divisors.
A300273 ranks collapsible partitions, counted by A275870.
A304442 counts partitions with all equal run-sums, ranked by A353833.
A333223 ranks knapsack compositions, counted by A325676.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353861 counts distinct partial run-sums of prime indices.
A354584 lists run-sums of prime indices, rows ranked by A353832.
Cf. A005811, A118914, A124010, A175413, A181819, A182857, A316413, A325862, A353834, A353835, A353836, A353931.

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

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

A354905 First position of n in A354578, where A354578(k) is the number of integer compositions whose run-sums constitute the k-th composition in standard order (graded reverse-lexicographic, A066099).

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A355748 Number of ways to choose a sequence of divisors, one of each part of the n-th composition in standard order.

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

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A354579 Number of distinct lengths of runs in the n-th composition in standard order.

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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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A354583 Heinz numbers of non-rucksack partitions: not every prime-power divisor has a different sum of prime indices.

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

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