A373954 -id:A373954 - cp's OEIS frontend

A374520 Numbers k such that the leaders of maximal anti-runs in the k-th composition in standard order (A066099) are not identical.

11, 19, 23, 26, 35, 39, 43, 46, 47, 53, 58, 67, 71, 74, 75, 78, 79, 83, 87, 91, 92, 93, 94, 95, 100, 106, 107, 117, 122, 131, 135, 138, 139, 142, 143, 147, 149, 151, 154, 155, 156, 157, 158, 159, 163, 164, 167, 171, 174, 175, 179, 183, 184, 185, 186, 187, 188

Offset: 1

Gus Wiseman, Aug 06 2024

nonn

The leaders of maximal anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

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 corresponding compositions begins:
  11: (2,1,1)
  19: (3,1,1)
  23: (2,1,1,1)
  26: (1,2,2)
  35: (4,1,1)
  39: (3,1,1,1)
  43: (2,2,1,1)
  46: (2,1,1,2)
  47: (2,1,1,1,1)
  53: (1,2,2,1)
  58: (1,1,2,2)
  67: (5,1,1)
  71: (4,1,1,1)
  74: (3,2,2)
  75: (3,2,1,1)
  78: (3,1,1,2)
  79: (3,1,1,1,1)
  83: (2,3,1,1)
  87: (2,2,1,1,1)
  91: (2,1,2,1,1)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For leaders of maximal constant runs we have the complement of A272919.
Positions of non-constant rows in A374515.
The complement is A374519, counted by A374517.
For distinct instead of identical leaders we have A374639, counted by A374678, complement A374638, counted by A374518.
Compositions of this type are counted by A374640.
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Six types of maximal runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A029931, A228351, A233564, A238343, A238424, A333213, A373949.

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

A373952 Number of integer compositions of n whose run-compression sums to 3.

Original entry on oeis.org

0, 0, 0, 3, 2, 4, 5, 6, 6, 9, 8, 10, 11, 12, 12, 15, 14, 16, 17, 18, 18, 21, 20, 22, 23, 24, 24, 27, 26, 28, 29, 30, 30, 33, 32, 34, 35, 36, 36, 39, 38, 40, 41, 42, 42, 45, 44, 46, 47, 48, 48, 51, 50, 52, 53, 54, 54, 57, 56, 58, 59, 60, 60, 63, 62, 64, 65, 66

Offset: 0

Gus Wiseman, Jun 29 2024

nonn

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 a(3) = 3 through a(9) = 9 compositions:
  (3)   (112)  (122)   (33)     (1222)    (11222)    (333)
  (12)  (211)  (221)   (1122)   (2221)    (22211)    (12222)
  (21)         (1112)  (2211)   (11122)   (111122)   (22221)
               (2111)  (11112)  (22111)   (221111)   (111222)
                       (21111)  (111112)  (1111112)  (222111)
                                (211111)  (2111111)  (1111122)
                                                     (2211111)
                                                     (11111112)
                                                     (21111111)

John Tyler Rascoe, Table of n, a(n) for n = 0..10000

For partitions we appear to have A137719.
Column k = 3 of A373949, rows-reversed A373951.
The compression-sum statistic is represented by A373953, difference A373954.
A003242 counts compressed compositions (anti-runs).
A011782 counts compositions.
A114901 counts compositions with no isolated parts.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A240085 counts compositions with no unique parts.
A333755 counts compositions by compressed length.
A373948 represents the run-compression transformation.
Cf. A037201 (halved A373947), A106356, A124762, A238130, A238279, A238343, A285981, A333213, A333489, A373950.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#]]==3&]],{n,0,10}]

A_x(N)={my(x='x+O('x^N)); concat([0, 0, 0], Vec(x^3 *(3-x-x^2-x^3)/((1-x)*(1-x^2)*(1-x^3))))}
A_x(50) \\ John Tyler Rascoe, Jul 01 2024

G.f.: x^3 * (3-x-x^2-x^3)/((1-x)*(1-x^2)*(1-x^3)). - John Tyler Rascoe, Jul 01 2024

a(26) onwards from John Tyler Rascoe, Jul 01 2024

A374250 Greatest sum of run-compression of a permutation of the prime factors of n.

Original entry on oeis.org

0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 7, 13, 9, 8, 2, 17, 8, 19, 9, 10, 13, 23, 7, 5, 15, 3, 11, 29, 10, 31, 2, 14, 19, 12, 10, 37, 21, 16, 9, 41, 12, 43, 15, 11, 25, 47, 7, 7, 12, 20, 17, 53, 8, 16, 11, 22, 31, 59, 12, 61, 33, 13, 2, 18, 16, 67, 21, 26, 14, 71

Offset: 1

Gus Wiseman, Jul 09 2024

nonn

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, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

			The prime factors of 24 are {2,2,2,3}, with permutations such as (2,2,3,2) whose run-compression sums to 7, so a(24) = 7.
The prime factors of 216 are {2,2,2,3,3,3}, with permutations such as (2,3,2,3,2,3) whose run-compression sums to 15, so a(216) = 15.

Positions of 2 are A000079 (powers of two) except 1.
Positions of 3 are A000244 (powers of three) except 1.
For least instead of greatest sum of run-compression we have A008472.
For prime indices instead of factors we have A373956.
For number of runs instead of sum of run-compression we have A373957.
A001221 counts distinct prime factors, A001222 with multiplicity.
A003242 counts run-compressed compositions, i.e., anti-runs.
A007947 (squarefree kernel) represents run-compression of multisets.
A008480 counts permutations of prime factors (or prime indices).
A056239 adds up prime indices, row sums of A112798.
A116861 counts partitions by sum of run-compression.
A304038 lists run-compression of prime indices, sum A066328.
A335433 lists numbers whose prime indices are separable, complement A335448.
A373949 counts compositions by sum of run-compression, opposite A373951.
A374251 run-compresses standard compositions, sum A373953, rank A373948.
Cf. A000040, A001223, A001414, A037201, A116608, A124767, A333755, A373954, A374246, A374247, A374252.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Max@@(Total[First/@Split[#]]& /@ Permutations[prifacs[n]]),{n,100}]

a(n) = A001414(n) iff n belongs to A335433 (the separable case, complement A335448), row-sums of A027746.

A373956 Greatest sum of run-compression of a permutation of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 4, 1, 2, 4, 5, 4, 6, 5, 5, 1, 7, 5, 8, 5, 6, 6, 9, 4, 3, 7, 2, 6, 10, 6, 11, 1, 7, 8, 7, 6, 12, 9, 8, 5, 13, 7, 14, 7, 7, 10, 15, 4, 4, 7, 9, 8, 16, 5, 8, 6, 10, 11, 17, 7, 18, 12, 8, 1, 9, 8, 19, 9, 11, 8, 20, 7, 21, 13, 8, 10, 9, 9, 22, 5

Offset: 1

Gus Wiseman, Jul 06 2024

nonn

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, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

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 prime indices of 24 are {1,1,1,2}, with permutations such as (1,1,2,1) whose run-compression sums to 4, so a(24) = 4.
The prime indices of 216 are {1,1,1,2,2,2}, with permutations such as (1,2,1,2,1,2) whose run-compression sums to 9, so a(216) = 9.

Positions of first appearances are 1 followed by the primes A000040.
Positions of 1 are A000079 (powers of two) except 1.
Positions of 2 are A000244 (powers of three) except 1.
Positions of 3 are {6} U A000351 (six or powers of five) except 1.
For number of runs instead of sum of run-compression we have A373957.
For prime factors instead of indices we have A374250.
A001221 counts distinct prime factors, A001222 with multiplicity.
A003242 counts run-compressed compositions, i.e., anti-runs.
A007947 (squarefree kernel) represents run-compression of multisets.
A008480 counts permutations of prime factors (or prime indices).
A056239 adds up prime indices, row sums of A112798.
A116861 counts partitions by sum of run-compression.
A304038 lists run-compression of prime indices, sum A066328.
A335433 lists numbers whose prime indices are separable, complement A335448.
A373949 counts compositions by sum of run-compression, opposite A373951.
A374251 run-compresses standard compositions, sum A373953, rank A373948.
Cf. A001223, A001414, A037201, A116608, A124767, A316413, A333755, A340387, A344415, A373954, A374246, A374247.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Max@@(Total[First/@Split[#]]&/@Permutations[prix[n]]),{n,100}]

a(n) = A056239(n) iff n belongs to A335433 (the separable case), complement A335448.

A374516 Sum of leaders of maximal anti-runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 31 2024

nonn

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

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 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1), with maximal anti-runs ((3,2,1,2),(2,1,2,5,1),(1),(1)), so a(1234567) is 3 + 2 + 1 + 1 = 7.

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For length instead of sum we have A333381.
Row-sums of A374515.
Other types of runs (instead of anti-):
- For identical runs we have A373953, row-sums of A374251.
- For weakly increasing runs we have A374630, row-sums of A374629.
- For strictly increasing runs we have A374684, row-sums of A374683.
- For weakly decreasing runs we have A374741, row-sums of A374740.
- For strictly decreasing runs we have A374758, row-sums of A374757.
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Ranks of non-contiguous compositions are A374253, counted by A335548.
Six types of maximal runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A029931, A228351, A233564, A238343, A272919, A373949, A374517, A374518, A374519, A374638.

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

A374248 Sum of prime indices of n (with multiplicity) minus the greatest possible sum of run-compression of a permutation of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 10 2024

nonn

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, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

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 prime indices of 96 are {1,1,1,1,1,2}, with sum 7, and we have permutations such as (1,1,1,1,2,1), with run-compression (1,2,1), with sum 4, so a(96) = 7 - 4 = 3.

Positions of zeros are A335433 (separable).
Positions of positive terms are A335448 (inseparable).
This is an opposite version of A373956, for prime factors A374250.
For prime factors instead of indices we have A374255.
A001221 counts distinct prime factors, A001222 with multiplicity.
A003242 counts run-compressed compositions, i.e., anti-runs.
A007947 (squarefree kernel) represents run-compression of multisets.
A008480 counts permutations of prime factors.
A027746 lists prime factors, row-sums A001414.
A027748 is run-compression of prime factors, row-sums A008472.
A056239 adds up prime indices, row-sums of A112798.
A116861 counts partitions by sum of run-compression.
A304038 is run-compression of prime indices, row-sums A066328.
A373949 counts compositions by sum of run-compression, opposite A373951.
A373957 gives greatest number of runs in a permutation of prime factors.
A374251 run-compresses standard compositions, sum A373953, rank A373948.
A374252 counts permutations of prime factors by number of runs.
Cf. A000040, A026549, A037201, A046660, A124767, A280286, A280292, A333755, A373954, A374246, A374247.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[prix[n]]-Max@@(Total[First/@Split[#]]&/@Permutations[prix[n]]),{n,100}]

a(n) = A056239(n) - A373956(n).

A374639 Numbers k such that the leaders of maximal anti-runs in the k-th composition in standard order (A066099) are not distinct.

Original entry on oeis.org

3, 7, 10, 14, 15, 21, 23, 27, 28, 29, 30, 31, 36, 39, 42, 43, 47, 51, 55, 56, 57, 58, 59, 60, 61, 62, 63, 71, 73, 79, 84, 85, 86, 87, 90, 94, 95, 99, 103, 106, 107, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 135

Offset: 1

Gus Wiseman, Aug 06 2024

nonn

The leaders of maximal anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

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 of terms together with the corresponding compositions begins:
   3: (1,1)
   7: (1,1,1)
  10: (2,2)
  14: (1,1,2)
  15: (1,1,1,1)
  21: (2,2,1)
  23: (2,1,1,1)
  27: (1,2,1,1)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  31: (1,1,1,1,1)

First differs from A335466 in lacking 166, complement A335467.
The complement for leaders of identical runs is A374249, counted by A274174.
For leaders of identical runs we have A374253, counted by A335548.
Positions of non-distinct (or non-strict) rows in A374515.
The complement is A374638, counted by A374518.
For identical instead of non-distinct we have A374519, counted by A374517.
For identical instead of distinct we have A374520, counted by A374640.
Compositions of this type are counted by A374678.
Other functional neighbors are A374768, A374698, A374701, A374767.
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
Six types of maximal runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A029931, A114994, A228351, A233564, A238343, A272919, A335466, A373949.

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

A375139 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are not weakly decreasing.

Original entry on oeis.org

26, 50, 53, 58, 90, 98, 100, 101, 106, 107, 114, 117, 122, 154, 164, 178, 181, 186, 194, 196, 197, 201, 202, 203, 210, 212, 213, 214, 215, 218, 226, 228, 229, 234, 235, 242, 245, 250, 282, 306, 309, 314, 324, 329, 346, 354, 356, 357, 362, 363, 370, 373, 378

Offset: 1

Gus Wiseman, Aug 12 2024

nonn

The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

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 corresponding compositions begin:
   26: (1,2,2)
   50: (1,3,2)
   53: (1,2,2,1)
   58: (1,1,2,2)
   90: (2,1,2,2)
   98: (1,4,2)
  100: (1,3,3)
  101: (1,3,2,1)
  106: (1,2,2,2)
  107: (1,2,2,1,1)
  114: (1,1,3,2)
  117: (1,1,2,2,1)
  122: (1,1,1,2,2)
  154: (3,1,2,2)
  164: (2,3,3)
  178: (2,1,3,2)
  181: (2,1,2,2,1)
  186: (2,1,1,2,2)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For leaders of identical runs we have A335485.
Ranked by positions of non-weakly decreasing rows in A374683.
For identical leaders we have A374685, counted by A374686.
The complement is counted by A374697.
For distinct leaders we have A374698, counted by A374687.
Compositions of this type are counted by A375135.
Weakly increasing leaders: A375137, counts A374636, complement A189076.
Interchanging weak/strict: A375295, counted by A375140, complement A188920.
A003242 counts anti-run compositions, ranks A333489.
A374700 counts compositions by sum of leaders of strictly increasing runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Leader is A065120.
- Parts are listed by A066099.
- Strict compositions are A233564.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Run-count: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Run-rank: A375123, A375124, A375125, A375126, A375127.
Cf. A000009, A261982, A333213, A374520, A374630, A374699, A374768.

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

A374254 Numbers k such that the k-th composition in standard order is an anti-run and matches the patterns (1,2,1) or (2,1,2).

Original entry on oeis.org

13, 22, 25, 45, 49, 54, 76, 77, 82, 89, 97, 101, 102, 105, 108, 109, 141, 148, 150, 153, 162, 165, 166, 177, 178, 180, 182, 193, 197, 198, 204, 205, 209, 210, 216, 217, 269, 278, 280, 281, 297, 300, 301, 305, 306, 308, 310, 322, 325, 326, 332, 333, 353, 354

Offset: 1

Gus Wiseman, Jul 14 2024

nonn

Such a composition cannot be strict.

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 standard compositions begin:
   13: (1,2,1)
   22: (2,1,2)
   25: (1,3,1)
   45: (2,1,2,1)
   49: (1,4,1)
   54: (1,2,1,2)
   76: (3,1,3)
   77: (3,1,2,1)
   82: (2,3,2)
   89: (2,1,3,1)
   97: (1,5,1)
  101: (1,3,2,1)
  102: (1,3,1,2)
  105: (1,2,3,1)
  108: (1,2,1,3)
  109: (1,2,1,2,1)
  141: (4,1,2,1)
  148: (3,2,3)
  150: (3,2,1,2)
  153: (3,1,3,1)

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

Compositions of this type are counted by A285981.
Permutations of prime indices of this type are counted by A335460.
This is the anti-run complement case of A374249, counted by A274174.
This is the anti-run case of A374253, counted by A335548.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A025047 counts wiggly compositions, ranks A345167.
A066099 lists compositions in standard order.
A124767 counts runs in standard compositions, anti-runs A333381.
A233564 ranks strict compositions, counted by A032020.
A333755 counts compositions by number of runs.
A335454 counts patterns matched by standard compositions.
A335456 counts patterns matched by compositions.
A335462 counts (1,2,1)- and (2,1,2)-matching permutations of prime indices.
A335465 counts minimal patterns avoided by a standard composition.
- A335470 counts (1,2,1)-matching compositions, ranks A335466.
- A335471 counts (1,2,1)-avoiding compositions, ranks A335467.
- A335472 counts (2,1,2)-matching compositions, ranks A335468.
- A335473 counts (2,1,2)-avoiding compositions, ranks A335469.
A373948 encodes run-compression using compositions in standard order.
A373949 counts compositions by run-compressed sum, opposite A373951.
A373953 gives run-compressed sum of standard compositions, excess A373954.
Cf. A106356, A124762, A238130, A238279, A261982, A333175, A333382, A333627, A335463, A335524, A335525.

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

Equals A333489 /\ A374253.

A373955 Numbers k such that the k-th integer composition in standard order contains two adjacent ones and no other runs.

Original entry on oeis.org

3, 11, 14, 19, 27, 28, 29, 35, 46, 51, 56, 57, 67, 75, 78, 83, 91, 92, 93, 99, 110, 112, 113, 114, 116, 118, 131, 139, 142, 155, 156, 157, 163, 179, 184, 185, 195, 203, 206, 211, 219, 220, 221, 224, 225, 226, 229, 230, 232, 233, 236, 237, 259, 267, 270, 275

Offset: 1

Gus Wiseman, Jun 29 2024

nonn

Also numbers k such that the excess compression of the k-th integer composition in standard order is 1.
The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
postn of 1 in

			The terms and corresponding compositions begin:
    3: (1,1)
   11: (2,1,1)
   14: (1,1,2)
   19: (3,1,1)
   27: (1,2,1,1)
   28: (1,1,3)
   29: (1,1,2,1)
   35: (4,1,1)
   46: (2,1,1,2)
   51: (1,3,1,1)
   56: (1,1,4)
   57: (1,1,3,1)
   67: (5,1,1)
   75: (3,2,1,1)
   78: (3,1,1,2)
   83: (2,3,1,1)
   91: (2,1,2,1,1)
   92: (2,1,1,3)
   93: (2,1,1,2,1)
   99: (1,4,1,1)

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

These compositions are counted by A373950.
Positions of ones in A373954.
A003242 counts compressed compositions (or anti-runs).
A114901 counts compositions with no isolated parts.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A240085 counts compositions with no unique parts.
A333755 counts compositions by compressed length.
A373948 encodes compression using compositions in standard order.
A373949 counts compositions by compression-sum.
A373953 gives compression-sum of standard compositions.
Cf. A106356, A124762, A238130, A238279, A238343, A285981, A333382, A333489, A373951, A373952.

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

cp's OEIS Frontend

A374520 Numbers k such that the leaders of maximal anti-runs in the k-th composition in standard order (A066099) are not identical.

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A373952 Number of integer compositions of n whose run-compression sums to 3.

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A374250 Greatest sum of run-compression of a permutation of the prime factors of n.

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A373956 Greatest sum of run-compression of a permutation of the prime indices of n.

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A374516 Sum of leaders of maximal anti-runs in the n-th composition in standard order.

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A374248 Sum of prime indices of n (with multiplicity) minus the greatest possible sum of run-compression of a permutation of the prime indices of n.

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A374639 Numbers k such that the leaders of maximal anti-runs in the k-th composition in standard order (A066099) are not distinct.

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A375139 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are not weakly decreasing.

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