A374249 -id:A374249 - cp's OEIS frontend

A376305 Run-compression of the sequence of first differences of squarefree numbers.

1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 2, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 4, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3

Offset: 1

Gus Wiseman, Sep 20 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 sequence of squarefree numbers (A005117) is:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ...
The sequence of first differences (A076259) of squarefree numbers is:
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, ...
The run-compression is A376305 (this sequence).

This is the run-compression of first differences of A005117.
For prime instead of squarefree numbers we have A037201, halved A373947.
Before compressing we had A076259, ones A375927.
For run-lengths instead of compression we have A376306.
For run-sums instead of compression we have A376307.
For prime-powers instead of squarefree numbers we have A376308.
For positions of first appearances instead of compression we have A376311.
The version for nonsquarefree numbers is A376312.
Positions of 1's are A376342.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A003242 counts compressed or anti-run compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A013929 lists nonsquarefree numbers, differences A078147.
A116861 counts partitions by compressed sum, by compressed length A116608.
A274174 counts contiguous compositions, ranks A374249.
Cf. A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A274174, A373197, A373198, A375707, A375708.

First/@Split[Differences[Select[Range[100],SquareFreeQ]]]

A374679 Number of integer compositions of n whose leaders of anti-runs are strictly increasing.

Original entry on oeis.org

1, 1, 1, 3, 4, 8, 15, 24, 45, 84, 142, 256, 464, 817, 1464, 2621, 4649, 8299, 14819, 26389, 47033, 83833, 149325, 266011, 473867, 843853

Offset: 0

Gus Wiseman, Aug 01 2024

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 a(0) = 1 through a(6) = 15 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)
                (12)  (13)   (14)   (15)
                (21)  (31)   (23)   (24)
                      (121)  (32)   (42)
                             (41)   (51)
                             (122)  (123)
                             (131)  (132)
                             (212)  (141)
                                    (213)
                                    (231)
                                    (312)
                                    (321)
                                    (1212)
                                    (1221)
                                    (2121)

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

For distinct but not necessarily increasing leaders we have A374518.
For partitions instead of compositions we have A375134.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we have A374634.
- For leaders of strictly increasing runs we have A374688.
- For leaders of strictly decreasing runs we have A374762.
Other types of run-leaders (instead of strictly increasing):
- For identical leaders we have A374517.
- For distinct leaders we have A374518.
- For weakly increasing leaders we have A374681.
- For weakly decreasing leaders we have A374682.
- For strictly decreasing leaders we have A374680.
A003242 counts anti-runs, ranks A333489.
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.
A274174 counts contiguous compositions, ranks A374249.
Cf. A188920, A238343, A333213, A333381, A373949, A374515, A374632, A374635, A374678, A374700, A374706, A375133.

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

A374744 Numbers k such that the leaders of weakly decreasing runs in the k-th composition in standard order (A066099) are identical.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 21, 22, 23, 31, 32, 33, 34, 35, 36, 37, 39, 42, 43, 45, 46, 47, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 76, 79, 85, 86, 87, 90, 91, 93, 94, 95, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138

Offset: 1

Gus Wiseman, Jul 24 2024

nonn

The leaders of weakly decreasing runs in a sequence are obtained by splitting into maximal weakly decreasing 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 the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   5: (2,1)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  10: (2,2)
  11: (2,1,1)
  15: (1,1,1,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  21: (2,2,1)
  22: (2,1,2)
  23: (2,1,1,1)
  31: (1,1,1,1,1)

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

Other types of runs and their counts: A272919 (A000005), A374519 (A374517), A374685 (A374686), A374759 (A374760).
The opposite is A374633, counted by A374631.
For distinct (instead of identical) leaders we have A374701, count A374743.
Positions of constant rows in A374740, opposite A374629, cf. A374630.
Compositions of this type are counted by A374742.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) (or sometimes A070939).
- Parts are listed by A066099.
- Adjacent equal pairs are counted by A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Cf. A065120, A106356, A238343, A333175, A333213, A335456, A373949, A374635, A374634, A374768.

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

A374748 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of weakly decreasing runs sum to k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 3, 2, 0, 1, 2, 6, 4, 3, 0, 1, 3, 9, 8, 7, 4, 0, 1, 3, 13, 15, 16, 11, 5, 0, 1, 4, 17, 24, 32, 28, 16, 6, 0, 1, 4, 23, 36, 58, 58, 44, 24, 8, 0, 1, 5, 28, 52, 96, 115, 100, 71, 34, 10, 0, 1, 5, 35, 72, 151, 203, 211, 176, 109, 49, 12

Offset: 0

Gus Wiseman, Jul 26 2024

The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   1   2
   0   1   2   3   2
   0   1   2   6   4   3
   0   1   3   9   8   7   4
   0   1   3  13  15  16  11   5
   0   1   4  17  24  32  28  16   6
   0   1   4  23  36  58  58  44  24   8
   0   1   5  28  52  96 115 100  71  34  10
   0   1   5  35  72 151 203 211 176 109  49  12
Row n = 6 counts the following compositions:
  .  (111111)  (222)    (33)     (42)    (51)    (6)
               (2211)   (321)    (411)   (141)   (15)
               (21111)  (3111)   (132)   (114)   (24)
                        (1221)   (1311)  (312)   (123)
                        (1122)   (1131)  (231)
                        (12111)  (1113)  (213)
                        (11211)  (2121)  (1212)
                        (11121)  (2112)
                        (11112)

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

Column n = k is A000009.
Column k = 2 is A004526.
Row-sums are A011782.
For length instead of sum we have A238343.
The opposite rank statistic is A374630, row-sums of A374629.
Column k = 3 is A374702.
The center n = 2k is A374703.
The corresponding rank statistic is A374741 row-sums of A374740.
Types of runs (instead of weakly decreasing):
- For leaders of constant runs we have A373949.
- For leaders of anti-runs we have A374521.
- For leaders of weakly increasing runs we have A374637.
- For leaders of strictly increasing runs we have A374700.
- For leaders of strictly decreasing runs we have A374766.
Types of run-leaders:
- For weakly increasing leaders we appear to have A188900.
- For identical leaders we have A374742, ranks A374744.
- For distinct leaders we have A374743, ranks A374701.
- For strictly decreasing leaders we have A374746.
- For weakly decreasing leaders we have A374747.
A003242 counts anti-run compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A335456 counts patterns matched by compositions.
A335548 counts non-contiguous compositions, ranks A374253.
Cf. A000041, A106356, A124766, A261982, A333213, A374251, A374761.

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

A375124 Weakly decreasing run-leader transformation for standard compositions.

Original entry on oeis.org

0, 1, 2, 1, 4, 2, 6, 1, 8, 4, 2, 2, 12, 6, 6, 1, 16, 8, 4, 4, 20, 2, 10, 2, 24, 12, 6, 6, 12, 6, 6, 1, 32, 16, 8, 8, 4, 4, 18, 4, 40, 20, 2, 2, 20, 10, 10, 2, 48, 24, 12, 12, 52, 6, 26, 6, 24, 12, 6, 6, 12, 6, 6, 1, 64, 32, 16, 16, 8, 8, 34, 8, 72, 4, 4, 4, 36

Offset: 0

Gus Wiseman, Aug 02 2024

nonn

The a(n)-th composition in standard order lists the leaders of weakly decreasing runs in the n-th composition in standard order.
The leaders of weakly decreasing runs in a sequence are obtained by splitting it into maximal weakly decreasing 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 813th composition in standard order is (1,3,2,1,2,1), with weakly decreasing runs ((1),(3,2,1),(2,1)), with leaders (1,3,2). This is the 50th composition in standard order, so a(813) = 50.

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

Positions of elements of A233564 are A374701, counted by A374743.
Positions of elements of A272919 are A374744, counted by A374742.
Ranks of rows of A374740.
The opposite version is A375123.
The strict version is A375126.
The strict opposite version is A375125.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) = A070939(n).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- 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.
- Run-sum transformation is A353847.
Six types of runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A065120, A106356, A189076, A238343, A333213, A373949, A374635, A374687, A374748, A374758.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[First/@Split[stc[n],GreaterEqual]],{n,0,100}]

A000120(a(n)) = A124765(n).
A065120(a(n)) = A065120(n).
A070939(a(n)) = A374741(n).

A375125 Strictly increasing run-leader transformation for standard compositions.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 1, 7, 8, 9, 10, 11, 1, 3, 3, 15, 16, 17, 18, 19, 2, 21, 5, 23, 1, 3, 6, 7, 3, 7, 7, 31, 32, 33, 34, 35, 36, 37, 9, 39, 2, 5, 42, 43, 5, 11, 11, 47, 1, 3, 6, 7, 1, 13, 3, 15, 3, 7, 14, 15, 7, 15, 15, 63, 64, 65, 66, 67, 68, 69, 17, 71, 4, 73

Offset: 0

Gus Wiseman, Aug 02 2024

nonn

The a(n)-th composition in standard order lists the leaders of strictly increasing runs in the n-th composition in standard order.
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 813th composition in standard order is (1,3,2,1,2,1), with strictly increasing runs ((1,3),(2),(1,2),(1)), with leaders (1,2,1,1). This is the 27th composition in standard order, so a(813) = 27.

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

Positions of elements of A233564 are A374698, counted by A374687.
Positions of elements of A272919 are A374685, counted by A374686.
Ranks of rows of A374683.
The weak version is A375123.
The weak opposite version is A375124.
The opposite version is A375126.
Other transformations: A375127, A373948.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) = A070939(n).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Run-sum transformation is A353847.
Six types of runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A065120, A106356, A189076, A238343, A333213, A373949, A374634, A374635, A374684, A374700, A375128.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[First/@Split[stc[n],Less]],{n,0,100}]

A000120(a(n)) = A124768(n).
A065120(a(n)) = A065120(n).
A070939(a(n)) = A374684(n).

A375126 Strictly decreasing run-leader transformation for standard compositions.

Original entry on oeis.org

0, 1, 2, 3, 4, 2, 6, 7, 8, 4, 10, 5, 12, 6, 14, 15, 16, 8, 4, 9, 20, 10, 10, 11, 24, 12, 26, 13, 28, 14, 30, 31, 32, 16, 8, 17, 36, 4, 18, 19, 40, 20, 42, 21, 20, 10, 22, 23, 48, 24, 12, 25, 52, 26, 26, 27, 56, 28, 58, 29, 60, 30, 62, 63, 64, 32, 16, 33, 8, 8

Offset: 0

Gus Wiseman, Aug 02 2024

nonn

The a(n)-th composition in standard order lists the leaders of strictly decreasing runs in the n-th composition in standard order.
The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing 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.
Does this sequence contain all nonnegative integers?

			The 813th composition in standard order is (1,3,2,1,2,1), with strictly decreasing runs ((1),(3,2,1),(2,1)), with leaders (1,3,2). This is the 50th composition in standard order, so a(813) = 50.

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

Positions of elements of A233564 are A374767, counted by A374761.
Positions of elements of A272919 are A374759, counted by A374760.
Ranks of rows of A374757 (row-sums A374758).
The weak opposite version is A375123.
The weak version is A375124.
The opposite version is A375125.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) = A070939(n).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Run-sum transformation is A353847.
Six types of runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A065120, A106356, A188920, A238343, A333213, A373949, A374634, A374685, A374744, A374766, A375128.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[First/@Split[stc[n],Greater]],{n,0,100}]

A000120(a(n)) = A124769(n).
A065120(a(n)) = A065120(n).
A070939(a(n)) = A374758(n).

A375127 The anti-run-leader transformation for standard compositions.

Original entry on oeis.org

0, 1, 2, 3, 4, 2, 1, 7, 8, 4, 10, 5, 1, 1, 3, 15, 16, 8, 4, 9, 2, 10, 2, 11, 1, 1, 6, 3, 3, 3, 7, 31, 32, 16, 8, 17, 36, 4, 4, 19, 2, 2, 42, 21, 2, 2, 5, 23, 1, 1, 1, 3, 1, 6, 1, 7, 3, 3, 14, 7, 7, 7, 15, 63, 64, 32, 16, 33, 8, 8, 8, 35, 4, 36, 18, 9, 4, 4, 9

Offset: 0

Gus Wiseman, Aug 02 2024

nonn

The a(n)-th composition in standard order lists the leaders of anti-runs of the n-th composition in standard order.
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.
Does this sequence contain all nonnegative integers?

			The 346th composition in standard order is (2,2,1,2,2), with anti-runs ((2),(2,1,2),(2)), with leaders (2,2,2). This is the 42nd composition in standard order, so a(346) = 42.

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

Positions of elements of A233564 are A374638, counted by A374518.
Positions of elements of A272919 are A374519, counted by A374517.
Ranks of rows of A374515.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) = A070939(n).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Run-sum transform is A353847.
Six types of runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A065120, A106356, A188920, A238343, A333213, A373949, A374516, A374521, A374685, A374698, A374701, A374767.

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

A000120(a(n)) = A333381(n).
A065120(a(n)) = A065120(n).
A070939(a(n)) = A374516(n).

A376306 Run-lengths of the sequence of first differences of squarefree numbers.

Original entry on oeis.org

2, 1, 2, 1, 1, 1, 2, 3, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 3, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 21 2024

nonn

			The sequence of squarefree numbers (A005117) is:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ...
The sequence of first differences (A076259) of squarefree numbers is:
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, ...
with runs:
  (1,1),(2),(1,1),(3),(1),(2),(1,1),(2,2,2),(1,1),(3,3),(1,1),(2),(1,1), ...
with lengths A376306 (this sequence).

Run-lengths of first differences of A005117.
Before taking run-lengths we had A076259, ones A375927.
For prime instead of squarefree numbers we have A333254.
For compression instead of run-lengths we have A376305.
For run-sums instead of run-lengths we have A376307.
For prime-powers instead of squarefree numbers we have A376309.
For positions of first appearances instead of run-lengths we have A376311.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed or anti-run compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A013929 lists nonsquarefree numbers, differences A078147.
A116861 counts partitions by compressed sum, by compressed length A116608.
A274174 counts contiguous compositions, ranks A374249.
Cf. A007674, A053797, A053806, A061398, A072284, A112925, A112926, A120992, A373198, A375707, A376312.

Length/@Split[Differences[Select[Range[100],SquareFreeQ]]]

A374636 Number of integer compositions of n whose leaders of maximal weakly increasing runs are not weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 3, 10, 28, 72, 178, 425, 985, 2237, 4999, 11016, 24006, 51822, 110983, 236064, 499168, 1050118, 2199304, 4587946, 9537506, 19765213, 40847186, 84205453, 173198096, 355520217, 728426569, 1489977348, 3043054678, 6206298312, 12641504738

Offset: 0

Gus Wiseman, Aug 09 2024

nonn

The leaders of maximal weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
Also the number of integer compositions of n matching the dashed pattern 1-32, ranked by A375137.
Also the number of integer compositions of n matching the dashed pattern 23-1, ranked by A375138.

			- The maximal weakly increasing runs of y = (1,1,3,2,1) are ((1,1,3),(2),(1)) with leaders (1,2,1) so y is counted under a(8). Also, y matches 1-32 and avoids 23-1.
- The maximal weakly increasing runs of y = (1,3,2,1,1) are ((1,3),(2),(1,1)) with leaders (1,2,1) so y is counted under a(8). Also, y matches 1-32 and avoids 23-1.
- The maximal weakly increasing runs of y = (2,3,1,1,1) are ((2,3),(1,1,1)) with leaders (2,1) so y is not counted under a(8). Also, y avoids 1-32 and matches 23-1.
- The maximal weakly increasing runs of y = (2,3,2,1) are ((2,3),(2),(1)) with leaders (2,2,1) so y is not counted under a(8). Also, y avoids 1-32 and matches 23-1.
- The maximal weakly increasing runs of y = (2,1,3,1,1) are ((2),(1,3),(1,1)) with leaders (2,1,1) so y is not counted under a(8). Also, y avoids both 1-32 and 23-1.
- The maximal weakly increasing runs of y = (2,1,1,3,1) are ((2),(1,1,3),(1)) with leaders (2,1,1) so y is not counted under a(8). Also, y avoids both 1-32 and 23-1.
The a(0) = 0 through a(8) = 10 compositions:
  .  .  .  .  .  .  (132)  (142)   (143)
                           (1132)  (152)
                           (1321)  (1142)
                                   (1232)
                                   (1322)
                                   (1421)
                                   (2132)
                                   (11132)
                                   (11321)
                                   (13211)

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

The reverse version is the same.
For leaders of identical runs we have A056823.
The complement is counted by A189076.
The non-dashed version is A335514.
For leaders of anti-runs we have A374699, complement A374682.
For weakly decreasing runs we have the complement of A374747.
For leaders of strictly increasing runs we have A375135, complement A374697.
These compositions are ranked by A375137, reverse A375138.
A003242 counts anti-runs, ranks A333489.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A274174 counts contiguous compositions, ranks A374249.
Cf. A000041, A188920, A238343, A238424, A333213, A373949, A374632, A374635, A374678, A374681, A375297.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],!GreaterEqual@@First/@Split[#,LessEqual]&]],{n,0,15}]
(* or *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{_,y_,z_,_,x_,_}/;x

a(n) = A011782(n) - A189076(n). - Jinyuan Wang, Feb 14 2025

More terms from Jinyuan Wang, Feb 14 2025

cp's OEIS Frontend

A376305 Run-compression of the sequence of first differences of squarefree numbers.

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A374679 Number of integer compositions of n whose leaders of anti-runs are strictly increasing.

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A374744 Numbers k such that the leaders of weakly decreasing runs in the k-th composition in standard order (A066099) are identical.

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A374748 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of weakly decreasing runs sum to k.

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A375124 Weakly decreasing run-leader transformation for standard compositions.

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A375125 Strictly increasing run-leader transformation for standard compositions.

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A375126 Strictly decreasing run-leader transformation for standard compositions.

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A375127 The anti-run-leader transformation for standard compositions.

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