A333489 -id:A333489 - cp's OEIS frontend

A353850 Number of integer compositions of n with all distinct run-sums.

1, 1, 2, 4, 5, 12, 24, 38, 52, 111, 218, 286, 520, 792, 1358, 2628, 4155, 5508, 9246, 13182, 23480, 45150, 54540, 94986, 146016, 213725, 301104, 478586, 851506, 1302234, 1775482, 2696942, 3746894, 6077784, 8194466, 12638334, 21763463, 28423976, 45309850, 62955524, 94345474

Offset: 0

Gus Wiseman, May 31 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 a(0) = 1 through a(5) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (1111)  (41)
                                (113)
                                (122)
                                (221)
                                (311)
                                (1112)
                                (2111)
                                (11111)
For n=4, (211) is invalid because the two runs (2) and (11) have the same sum. - _Joseph Likar_, Aug 04 2023

Joseph Likar, Table of n, a(n) for n = 0..120

For distinct parts instead of run-sums we have A032020.
For distinct multiplicities instead of run-sums we have A242882.
For distinct run-lengths instead of run-sums we have A329739, ptns A098859.
For runs instead of run-sums we have A351013.
For partitions we have A353837, ranked by A353838 (complement A353839).
For equal instead of distinct run-sums we have A353851, ptns A304442.
These compositions are ranked by A353852.
The weak version (rucksack compositions) is A354580, ranked by A354581.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A175413 lists numbers whose binary expansion has all distinct runs.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353847 gives composition run-sum transformation.
A353929 counts distinct runs in binary expansion, firsts A353930.
Cf. A238279, A333755, A351016, A351017, A353832, A353848, A353849, A353853-A353859, A353860, A353863, A353932.

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

Terms a(21) and onwards from Joseph Likar, Aug 04 2023

A373953 Sum of run-compression of the n-th integer composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 25 2024

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 the (run-) compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has compression (1,2,1).

			The standard compositions and their compressions and compression sums begin:
   0: ()        --> ()      --> 0
   1: (1)       --> (1)     --> 1
   2: (2)       --> (2)     --> 2
   3: (1,1)     --> (1)     --> 1
   4: (3)       --> (3)     --> 3
   5: (2,1)     --> (2,1)   --> 3
   6: (1,2)     --> (1,2)   --> 3
   7: (1,1,1)   --> (1)     --> 1
   8: (4)       --> (4)     --> 4
   9: (3,1)     --> (3,1)   --> 4
  10: (2,2)     --> (2)     --> 2
  11: (2,1,1)   --> (2,1)   --> 3
  12: (1,3)     --> (1,3)   --> 4
  13: (1,2,1)   --> (1,2,1) --> 4
  14: (1,1,2)   --> (1,2)   --> 3
  15: (1,1,1,1) --> (1)     --> 1

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Positions of 1's are A000225.
Counting partitions by this statistic gives A116861, by length A116608.
For length instead of sum we have A124767, counted by A238279 and A333755.
Compositions counted by this statistic are A373949, opposite A373951.
A037201 gives compression of first differences of primes, halved A373947.
A066099 lists the parts of all compositions in standard order.
A114901 counts compositions with no isolated parts.
A240085 counts compositions with no unique parts.
A333489 ranks anti-runs, counted by A003242.
Cf. A106356, A124762, A238130, A238343, A272919, A285981, A333381, A333382, A333627, A373952, A373954.

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

a(n) = A029837(A373948(n)).

A353848 Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.

Original entry on oeis.org

0, 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, 143, 168, 170, 175, 187, 238, 248, 250, 255, 256, 292, 316, 487, 511, 512, 528, 543, 682, 750, 955, 1008, 1023, 1024, 2047, 2048, 2080, 2084, 2090, 2111, 2184

Offset: 0

Gus Wiseman, May 30 2022

nonn

Every sequence can be uniquely split into non-overlapping runs, 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 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:
     0:       0  ()
     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)
For example:
- The 59th composition in standard order is (1,1,2,1,1), with run-sums (2,2,2), so 59 is in the sequence.
- The 2298th composition in standard order is (4,1,1,1,1,2,2), with run-sums (4,4,4), so 2298 is in the sequence.
- The 2346th composition in standard order is (3,3,2,2,2), with run-sums (6,6), so 2346 is in the sequence.

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

Standard compositions are listed by A066099.
For equal lengths instead of sums we have A353744, counted by A329738.
The version for partitions is A353833, counted by A304442.
These compositions are counted by A353851.
The distinct instead of equal version is A353852, counted by A353850.
The run-sums themselves are listed by A353932, with A353849 distinct terms.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353840-A353846 pertain to partition run-sum trajectory.
A353847 represents the run-sum transformation for compositions.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
A353863 counts run-sum-complete partitions.
Cf. A003242, A047966, A106356, A140690, A238279, A274174, A333381, A333489, A333755, A353832, A353864.

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

A353849(a(n)) = 1.

A175413 Those positive integers n that when written in binary, the lengths of the runs of 1 are distinct and the lengths of the runs of 0's are distinct.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 15, 16, 19, 23, 24, 25, 28, 29, 30, 31, 32, 35, 38, 39, 44, 47, 48, 49, 50, 52, 55, 56, 57, 59, 60, 61, 62, 63, 64, 67, 70, 71, 78, 79, 88, 92, 95, 96, 97, 98, 103, 104, 111, 112, 113, 114, 115, 116, 120, 121, 123, 124, 125

Offset: 1

Leroy Quet, May 07 2010

A044813 contains those positive integers that when written in binary, have all run-lengths (of both 0's and 1's) distinct.
A175414 contains those positive integers in A175413 that are not in A044813. (A175414 contains those positive integers that when written in binary, at least one run of 0's is the same length as one run of 1's, even though all run of 0 are of distinct length and all runs of 1's are of distinct length.)
Also numbers whose binary expansion has all distinct runs (not necessarily run-lengths). - Gus Wiseman, Feb 21 2022

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A044813, A175414.
Runs in binary expansion are counted by A005811, distinct A297770.
The complement is A351205.
The version for standard compositions is A351290, complement A351291.
A000120 counts binary weight.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A325545 counts compositions with distinct differences.
A333489 ranks anti-runs, complement A348612, counted by A003242.
A334028 counts distinct parts in standard compositions.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A070939, A085207, A098859, A233564, A238130 or A238279, A283353, A328592, A350952, A351015, A351203.

q:= proc(n) uses ListTools; (l-> is(nops(l)=add(
      nops(i), i={Split(`=`, l, 1)}) +add(
      nops(i), i={Split(`=`, l, 0)})))(Bits[Split](n))
    end:
select(q, [$1..200])[];  # Alois P. Heinz, Mar 14 2022

f[n_] := And@@Unequal@@@Transpose[Partition[Length/@Split[IntegerDigits[n, 2]], 2, 2, {1,1}, 0]]; Select[Range[125], f] (* Ray Chandler, Oct 21 2011 *)
Select[Range[0,100],UnsameQ@@Split[IntegerDigits[#,2]]&] (* Gus Wiseman, Feb 21 2022 *)

from itertools import groupby, product
def ok(n):
    runs = [(k, len(list(g))) for k, g in groupby(bin(n)[2:])]
    return len(runs) == len(set(runs))
print([k for k in range(1, 125) if ok(k)]) # Michael S. Branicky, Feb 22 2022

Extended by Ray Chandler, Oct 21 2011

A335126 A multiset whose multiplicities are the prime indices of n is inseparable.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 28, 29, 31, 33, 34, 37, 38, 39, 41, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 97, 101, 102, 103, 104, 106

Offset: 1

Gus Wiseman, Jul 01 2020

nonn

A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

			The sequence of terms together with the corresponding multisets begins:
   3: {1,1}
   5: {1,1,1}
   7: {1,1,1,1}
  10: {1,1,1,2}
  11: {1,1,1,1,1}
  13: {1,1,1,1,1,1}
  14: {1,1,1,1,2}
  17: {1,1,1,1,1,1,1}
  19: {1,1,1,1,1,1,1,1}
  21: {1,1,1,1,2,2}
  22: {1,1,1,1,1,2}
  23: {1,1,1,1,1,1,1,1,1}
  26: {1,1,1,1,1,1,2}
  28: {1,1,1,1,2,3}
  29: {1,1,1,1,1,1,1,1,1,1}

The complement is A335127.
Anti-run compositions are A003242.
Anti-runs are ranked by A333489.
Separable partitions are A325534.
Inseparable partitions are A325535.
Separable factorizations are A335434.
Inseparable factorizations are A333487.
Separable partitions are ranked by A335433.
Inseparable partitions are ranked by A335448.
Anti-run permutations of prime indices are A335452.
Patterns contiguously matched by compositions are A335457.
Cf. A025487, A056239, A106351, A112798, A114938, A181819, A181821, A278990, A292884, A335407, A335489, A335516, A335838.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Select[Permutations[nrmptn[#]],!MatchQ[#,{_,x_,x_,_}]&]=={}&]

A345164 Number of alternating permutations of the multiset of prime factors of n.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 4, 1, 0, 2, 2, 2, 2, 1, 2, 2, 0, 1, 4, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 4, 1, 2, 1, 0, 2, 4, 1, 1, 2, 4, 1, 1, 1, 2, 1, 1, 2, 4, 1, 0, 0, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Jun 13 2021

nonn

First differs from A335452 at a(30) = 4, A335452(30) = 6. The anti-runs (2,3,5) and (5,3,2) are not alternating.

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

			The a(n) alternating permutations of prime indices for n = 180, 210, 300, 420, 900:
  (12132)  (1324)  (13132)  (12143)  (121323)
  (21213)  (1423)  (13231)  (13142)  (132312)
  (21312)  (2143)  (21313)  (13241)  (213132)
  (23121)  (2314)  (23131)  (14132)  (213231)
  (31212)  (2413)  (31213)  (14231)  (231213)
           (3142)  (31312)  (21314)  (231312)
           (3241)           (21413)  (312132)
           (3412)           (23141)  (323121)
           (4132)           (24131)
           (4231)           (31214)
                            (31412)
                            (34121)
                            (41213)
                            (41312)

Counting all permutations gives A008480.
Dominated by A335452 (number of separations of prime factors).
Including twins (x,x) gives A344606.
Positions of zeros are A345171, counted by A345165.
Positions of nonzero terms are A345172.
A000041 counts integer partitions.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A025047 counts alternating or wiggly compositions, also A025048, A025049.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A344654 counts non-twin partitions w/o alternating permutation, rank: A344653.
A344740 counts twins and partitions w/ alternating permutation, rank: A344742.
A345166 counts separable partitions w/o alternating permutation, rank: A345173.
A345170 counts partitions with a alternating permutation.
Cf. A001222, A071321, A071322, A316523, A316524, A333489, A335126, A344605, A344614, A344616, A344652, A345163, A345168.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]],wigQ]],{n,30}]

A353932 Irregular triangle read by rows where row k lists the run-sums of the k-th composition in standard order.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jun 10 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 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 1
  1 2
  3
  4
  3 1
  4
  2 2
  1 3
  1 2 1
For example, composition 350 in standard order is (2,2,1,1,1,2), so row 350 is (4,3,2).

Row-sums are A029837.
Standard compositions are listed by A066099.
Row-lengths are A124767.
These compositions are ranked by A353847.
Row k has A353849(k) distinct parts.
The version for partitions is A354584, ranked by A353832.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
Cf. A003242, A175413, A181819, A238279, A274174, A333381, A333489, A333755, A353835, A353839, A353863, A353864.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Total/@Split[stc[n]],{n,0,30}]

A373954 Excess run-compression of standard compositions. Sum of all parts minus sum of compressed parts of the n-th integer composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 27 2024

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 the (run-) compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has compression (1,2,1).

			The excess compression of (2,1,1,3) is 1, so a(92) = 1.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

For length instead of sum we have A124762, counted by A106356.
The opposite for length is A124767, counted by A238279 and A333755.
Positions of zeros are A333489, counted by A003242.
Positions of nonzeros are A348612, counted by A131044.
Compositions counted by this statistic are A373951, opposite A373949.
Compression of standard compositions is A373953.
Positions of ones are A373955.
A037201 gives compression of first differences of primes, halved A373947.
A066099 lists the parts of all compositions in standard order.
A114901 counts compositions with no isolated parts.
A116861 counts partitions by this statistic, by length A116608.
A240085 counts compositions with no unique parts.
A333627 takes the rank of a composition to the rank of its run-lengths.
Cf. A070939, A238130, A238343, A272919, A285981, A333213, A333381, A333382.

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

a(n) = A029837(n) - A373953(n).

A353852 Numbers k such that the k-th composition in standard order (row k of A066099) has all distinct run-sums.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 47, 48, 50, 51, 52, 55, 56, 57, 58, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 79, 80, 81, 84, 85, 86, 87, 88

Offset: 0

Gus Wiseman, May 31 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 together with their binary expansions and corresponding compositions begin:
   0:        0  ()
   1:        1  (1)
   2:       10  (2)
   3:       11  (1,1)
   4:      100  (3)
   5:      101  (2,1)
   6:      110  (1,2)
   7:      111  (1,1,1)
   8:     1000  (4)
   9:     1001  (3,1)
  10:     1010  (2,2)
  12:     1100  (1,3)
  15:     1111  (1,1,1,1)
  16:    10000  (5)
  17:    10001  (4,1)
  18:    10010  (3,2)
  19:    10011  (3,1,1)
  20:    10100  (2,3)
  21:    10101  (2,2,1)
  23:    10111  (2,1,1,1)

The version for runs in binary expansion is A175413.
The version for parts instead of run-sums is A233564, counted A032020.
The version for run-lengths instead of run-sums is A351596, counted A329739.
The version for runs instead of run-sums is A351290, counted by A351013.
The version for partitions is A353838, counted A353837, complement A353839.
The equal instead of distinct version is A353848, counted by A353851.
These compositions are counted by A353850.
The weak version (rucksack compositions) is A354581, counted by A354580.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A242882 counts composition with distinct multiplicities, partitions A098859.
A304442 counts partitions with all equal run-sums.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353853-A353859 pertain to composition run-sum trajectory.
A353864 counts rucksack partitions, perfect A353865.
A353929 counts distinct runs in binary expansion, firsts A353930.
Cf. A044813, A238279, A333755, A351016, A351017, A353832, A353847, A353849, A353860, A353863, A353932.

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

A357136 Triangle read by rows where T(n,k) is the number of integer compositions of n with alternating sum k = 0..n. Part of the full triangle A097805.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 3, 0, 3, 0, 1, 0, 6, 0, 4, 0, 1, 10, 0, 10, 0, 5, 0, 1, 0, 20, 0, 15, 0, 6, 0, 1, 35, 0, 35, 0, 21, 0, 7, 0, 1, 0, 70, 0, 56, 0, 28, 0, 8, 0, 1, 126, 0, 126, 0, 84, 0, 36, 0, 9, 0, 1, 0, 252, 0, 210, 0, 120, 0, 45, 0, 10, 0, 1

Offset: 0

Gus Wiseman, Sep 30 2022

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

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

			Triangle begins:
    1
    0   1
    1   0   1
    0   2   0   1
    3   0   3   0   1
    0   6   0   4   0   1
   10   0  10   0   5   0   1
    0  20   0  15   0   6   0   1
   35   0  35   0  21   0   7   0   1
    0  70   0  56   0  28   0   8   0   1
  126   0 126   0  84   0  36   0   9   0   1
    0 252   0 210   0 120   0  45   0  10   0   1
  462   0 462   0 330   0 165   0  55   0  11   0   1
    0 924   0 792   0 495   0 220   0  66   0  12   0   1
For example, row n = 5 counts the following compositions:
  .  (32)     .  (41)   .  (5)
     (122)       (113)
     (221)       (212)
     (1121)      (311)
     (2111)
     (11111)

The full triangle counting compositions by alternating sum is A097805.
The version for partitions is A103919, full triangle A344651.
This is the right-half of even-indexed rows of A260492.
The triangle without top row and left column is A108044.
Ranking and counting compositions:
- product = sum: A335404, counted by A335405.
- sum = twice alternating sum: A348614, counted by A262977.
- length = alternating sum: A357184, counted by A357182.
- length = absolute value of alternating sum: A357185, counted by A357183.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A025047 counts alternating compositions, ranked by A345167.
A032020 counts strict compositions, ranked by A233564.
A124754 gives alternating sums of standard compositions.
A238279 counts compositions by sum and number of maximal runs.
Cf. A000120, A051159, A070939, A114220, A114901, A242882, A262046.

Prepend[Table[If[EvenQ[nn],Prepend[#,0],#]&[Riffle[Table[Binomial[nn,k],{k,Floor[nn/2],nn}],0]],{nn,0,10}],{1}]

cp's OEIS Frontend

A353850 Number of integer compositions of n with all distinct run-sums.

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A373953 Sum of run-compression of the n-th integer composition in standard order.

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A353848 Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.

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A175413 Those positive integers n that when written in binary, the lengths of the runs of 1 are distinct and the lengths of the runs of 0's are distinct.

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A335126 A multiset whose multiplicities are the prime indices of n is inseparable.

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A345164 Number of alternating permutations of the multiset of prime factors of n.

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A353932 Irregular triangle read by rows where row k lists the run-sums of the k-th composition in standard order.

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A373954 Excess run-compression of standard compositions. Sum of all parts minus sum of compressed parts of the n-th integer composition in standard order.

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