cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 34 results. Next

A374629 Irregular triangle listing the leaders of maximal weakly increasing runs in the n-th composition in standard order.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jul 20 2024

Keywords

Comments

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.
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.

Examples

			The 58654th composition in standard order is (1,1,3,2,4,1,1,1,2), with maximal weakly increasing runs ((1,1,3),(2,4),(1,1,1,2)), so row 58654 is (1,2,1).
The nonnegative integers, corresponding compositions, and leaders of maximal weakly increasing runs begin:
    0:      () -> ()      15: (1,1,1,1) -> (1)
    1:     (1) -> (1)     16:       (5) -> (5)
    2:     (2) -> (2)     17:     (4,1) -> (4,1)
    3:   (1,1) -> (1)     18:     (3,2) -> (3,2)
    4:     (3) -> (3)     19:   (3,1,1) -> (3,1)
    5:   (2,1) -> (2,1)   20:     (2,3) -> (2)
    6:   (1,2) -> (1)     21:   (2,2,1) -> (2,1)
    7: (1,1,1) -> (1)     22:   (2,1,2) -> (2,1)
    8:     (4) -> (4)     23: (2,1,1,1) -> (2,1)
    9:   (3,1) -> (3,1)   24:     (1,4) -> (1)
   10:   (2,2) -> (2)     25:   (1,3,1) -> (1,1)
   11: (2,1,1) -> (2,1)   26:   (1,2,2) -> (1)
   12:   (1,3) -> (1)     27: (1,2,1,1) -> (1,1)
   13: (1,2,1) -> (1,1)   28:   (1,1,3) -> (1)
   14: (1,1,2) -> (1)     29: (1,1,2,1) -> (1,1)

Links

Crossrefs

Row-leaders are A065120.
Row-lengths are A124766.
Row-sums are A374630.
Positions of constant rows are A374633, counted by A374631.
Positions of strict rows are A374768, counted by A374632.
For other types of runs we have A374251, A374515, A374683, A374740, A374757.
Positions of non-weakly decreasing rows are A375137.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
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, reverse A228351.
- Number of adjacent equal pairs is 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, length A124767, 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.
Cf. A046660, A106356, A188920, A189076, A238343, A272919, A333213, A373949, A374634, A374635, A374637, A374701, A375123.

Programs

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

A374683 Irregular triangle read by rows where row n lists the leaders of strictly increasing runs in the n-th composition in standard order.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jul 26 2024

Keywords

Comments

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.

Examples

			The maximal strictly increasing subsequences of the 1234567th composition in standard order are ((3),(2),(1,2),(2),(1,2,5),(1),(1),(1)), so row 1234567 is (3,2,1,2,1,1,1,1).
The nonnegative integers, corresponding compositions, and leaders of strictly increasing runs begin:
   0:      () -> ()         15: (1,1,1,1) -> (1,1,1,1)
   1:     (1) -> (1)        16:       (5) -> (5)
   2:     (2) -> (2)        17:     (4,1) -> (4,1)
   3:   (1,1) -> (1,1)      18:     (3,2) -> (3,2)
   4:     (3) -> (3)        19:   (3,1,1) -> (3,1,1)
   5:   (2,1) -> (2,1)      20:     (2,3) -> (2)
   6:   (1,2) -> (1)        21:   (2,2,1) -> (2,2,1)
   7: (1,1,1) -> (1,1,1)    22:   (2,1,2) -> (2,1)
   8:     (4) -> (4)        23: (2,1,1,1) -> (2,1,1,1)
   9:   (3,1) -> (3,1)      24:     (1,4) -> (1)
  10:   (2,2) -> (2,2)      25:   (1,3,1) -> (1,1)
  11: (2,1,1) -> (2,1,1)    26:   (1,2,2) -> (1,2)
  12:   (1,3) -> (1)        27: (1,2,1,1) -> (1,1,1)
  13: (1,2,1) -> (1,1)      28:   (1,1,3) -> (1,1)
  14: (1,1,2) -> (1,1)      29: (1,1,2,1) -> (1,1,1)

Links

Crossrefs

Row-leaders are A065120.
Row-lengths are A124768.
Other types of runs: A374251, A374515, A374740.
The weak version is A374629, sum A374630, length A124766.
Row-sums are A374684.
Positions of identical rows are A374685, counted by A374686.
Positions of distinct (strict) rows are A374698, counted by A374687.
The opposite version is A374757, sum A374758, length A124769.
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.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124767, A333381.
- 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.
Cf. A106356, A188920, A189076, A238343, A272919, A333213, A373949, A374700.

Programs

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

A374740 Irregular triangle read by rows where row n lists the leaders of weakly decreasing runs in the n-th composition in standard order.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jul 24 2024

Keywords

Comments

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.

Examples

			The maximal weakly decreasing subsequences of the 1234567th composition in standard order are ((3,2,1),(2,2,1),(2),(5,1,1,1)), so row 1234567 is (3,2,2,5).
The nonnegative integers, corresponding compositions, and leaders of weakly decreasing runs begin:
    0: () -> ()           15: (1,1,1,1) -> (1)
    1: (1) -> (1)         16: (5) -> (5)
    2: (2) -> (2)         17: (4,1) -> (4)
    3: (1,1) -> (1)       18: (3,2) -> (3)
    4: (3) -> (3)         19: (3,1,1) -> (3)
    5: (2,1) -> (2)       20: (2,3) -> (2,3)
    6: (1,2) -> (1,2)     21: (2,2,1) -> (2)
    7: (1,1,1) -> (1)     22: (2,1,2) -> (2,2)
    8: (4) -> (4)         23: (2,1,1,1) -> (2)
    9: (3,1) -> (3)       24: (1,4) -> (1,4)
   10: (2,2) -> (2)       25: (1,3,1) -> (1,3)
   11: (2,1,1) -> (2)     26: (1,2,2) -> (1,2)
   12: (1,3) -> (1,3)     27: (1,2,1,1) -> (1,2)
   13: (1,2,1) -> (1,2)   28: (1,1,3) -> (1,3)
   14: (1,1,2) -> (1,2)   29: (1,1,2,1) -> (1,2)

Links

Crossrefs

Row-leaders are A065120.
Row-lengths are A124765.
Other types of runs are A374251, A374515, A374683, A374757.
The opposite is A374629.
Positions of distinct (strict) rows are A374701, counted by A374743.
Row-sums are A374741, opposite A374630.
Positions of identical rows are A374744, counted by A374742.
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.
- Number of adjacent equal pairs is 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, 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.
Cf. A106356, A188920, A189076, A238343, A272919, A333213, A373949, A374634, A374635, A374637.

Programs

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

A374518 Number of integer compositions of n whose leaders of anti-runs are distinct.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 17, 32, 58, 112, 201, 371, 694, 1276, 2342, 4330, 7958, 14613, 26866, 49303, 90369, 165646, 303342, 555056, 1015069, 1855230
Offset: 0

Views

Author

Gus Wiseman, Aug 01 2024

Keywords

Comments

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.

Examples

			The a(0) = 1 through a(6) = 17 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)
                (12)  (13)   (14)   (15)
                (21)  (31)   (23)   (24)
                      (121)  (32)   (42)
                      (211)  (41)   (51)
                             (122)  (123)
                             (131)  (132)
                             (212)  (141)
                             (311)  (213)
                                    (231)
                                    (312)
                                    (321)
                                    (411)
                                    (1212)
                                    (1221)
                                    (2112)
                                    (2121)

Links

Crossrefs

These compositions have ranks A374638.
The complement is counted by A374678.
For partitions instead of compositions we have A375133.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A274174, ranks A374249.
- For leaders of weakly increasing runs we have A374632, ranks A374768.
- For leaders of strictly increasing runs we have A374687, ranks A374698.
- For leaders of weakly decreasing runs we have A374743, ranks A374701.
- For leaders of strictly decreasing runs we have A374761, ranks A374767.
Other types of run-leaders (instead of distinct):
- For identical leaders we have A374517.
- For weakly increasing leaders we have A374681.
- For strictly increasing leaders we have A374679.
- 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.
Cf. A188920, A233564, A238343, A333213, A333381, A373949, A374515.

Programs

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

A374517 Number of integer compositions of n whose leaders of anti-runs are identical.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 25, 46, 85, 160, 301, 561, 1056, 1984, 3730, 7037, 13273, 25056, 47382, 89666, 169833, 322038, 611128, 1160660, 2206219, 4196730, 7988731, 15217557, 29005987, 55321015, 105570219, 201569648, 385059094, 735929616, 1407145439, 2691681402
Offset: 0

Views

Author

Gus Wiseman, Aug 01 2024

Keywords

Comments

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.

Examples

			The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (1111)  (131)
                                (212)
                                (221)
                                (1112)
                                (1121)
                                (1211)
                                (11111)

Links

Crossrefs

For partitions instead of compositions we have A034296 or A115029.
These compositions have ranks A374519.
The complement is counted by A374640.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A000005 for n > 0, ranks A272919.
- For leaders of weakly increasing runs we have A374631, ranks A374633.
- For leaders of strictly increasing runs we have A374686, ranks A374685.
- For leaders of weakly decreasing runs we have A374742, ranks A374741.
- For leaders of strictly decreasing runs we have A374760, ranks A374759.
Other types of run-leaders (instead of identical):
- For distinct leaders we have A374518.
- For weakly increasing leaders we have A374681.
- For strictly increasing leaders we have A374679.
- 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, A189076, A238343, A333213, A373949, A374515, A374518.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],SameQ@@First/@Split[#,UnsameQ]&]],{n,0,15}]
  • PARI
    C_x(N) = {my(g =1/(1 - sum(k=1, N, x^k/(1+x^k))));g}
A_x(i,N) = {my(x='x+O('x^N), f=(x^i)*(C_x(N)*(x^i)+x^i+1)/(1+x^i)^2);f}
B_x(i,j,N) = {my(x='x+O('x^N), f=C_x(N)*x^(i+j)/((1+x^i)*(1+x^j)));f}
D_x(N) = {my(x='x+O('x^N), f=1+sum(i=1,N,-1+sum(j=0,N-i, A_x(i,N)^j)*(1-B_x(i,i,N)+sum(k=1,N-i,B_x(i,k,N)))));Vec(f)}
D_x(30) \\ John Tyler Rascoe, Aug 16 2024

Formula

G.f.: 1 + Sum_{i>0} (-1 + Sum_{j>=0} (A(i,x)^j)*(1 + Sum_{k>0, k<>i} (B(i,k,x)))) where A(i,x) = (x^i)*(C(x)*(x^i) + x^i + 1)/(1+x^i)^2, B(i,k,x) = C(x)*x^(i+k)/((1+x^i)*(1+x^k)), and C(x) is the g.f. for A003242. - John Tyler Rascoe, Aug 16 2024

Extensions

a(26) onwards from John Tyler Rascoe, Aug 16 2024

A374757 Irregular triangle read by rows where row n lists the leaders of strictly decreasing runs in the n-th composition in standard order.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jul 29 2024

Keywords

Comments

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.

Examples

			the 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1), with strictly decreasing runs ((3,2,1),(2),(2,1),(2),(5,1),(1),(1)), so row 1234567 is (3,2,2,2,5,1,1).
The nonnegative integers, corresponding compositions, and leaders of strictly decreasing runs begin:
    0:      () -> ()        15: (1,1,1,1) -> (1,1,1,1)
    1:     (1) -> (1)       16:       (5) -> (5)
    2:     (2) -> (2)       17:     (4,1) -> (4)
    3:   (1,1) -> (1,1)     18:     (3,2) -> (3)
    4:     (3) -> (3)       19:   (3,1,1) -> (3,1)
    5:   (2,1) -> (2)       20:     (2,3) -> (2,3)
    6:   (1,2) -> (1,2)     21:   (2,2,1) -> (2,2)
    7: (1,1,1) -> (1,1,1)   22:   (2,1,2) -> (2,2)
    8:     (4) -> (4)       23: (2,1,1,1) -> (2,1,1)
    9:   (3,1) -> (3)       24:     (1,4) -> (1,4)
   10:   (2,2) -> (2,2)     25:   (1,3,1) -> (1,3)
   11: (2,1,1) -> (2,1)     26:   (1,2,2) -> (1,2,2)
   12:   (1,3) -> (1,3)     27: (1,2,1,1) -> (1,2,1)
   13: (1,2,1) -> (1,2)     28:   (1,1,3) -> (1,1,3)
   14: (1,1,2) -> (1,1,2)   29: (1,1,2,1) -> (1,1,2)

Links

Crossrefs

Row-leaders of nonempty rows are A065120.
Row-lengths are A124769.
The opposite version is A374683, sum A374684, length A124768.
The weak version is A374740, sum A374741, length A124765.
Row-sums are A374758.
Positions of identical rows are A374759 (counted by A374760).
Positions of distinct (strict) rows are A374767 (counted by A374761).
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.
- 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 runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A051903, A106356, A188920, A189076, A233564, A238343, A333213, A373949, A374685, A374698, A374700, A374706.

Programs

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

A374767 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are distinct.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 24, 25, 32, 33, 34, 35, 37, 38, 40, 41, 44, 48, 49, 50, 52, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 77, 78, 80, 81, 82, 83, 88, 89, 92, 96, 97, 98, 101, 102, 104, 105, 108, 128, 129, 130, 131, 132, 133
Offset: 1

Views

Author

Gus Wiseman, Jul 29 2024

Keywords

Comments

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.

Examples

			The 10000000th composition in standard order is (3,1,4,3,2,1,2,8), with strictly decreasing runs ((3,1),(4,3,2,1),(2),(8)), with leaders (3,4,2,1) so 10000000 is in the sequence.
The terms together with the corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   6: (1,2)
   8: (4)
   9: (3,1)
  11: (2,1,1)
  12: (1,3)
  13: (1,2,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  20: (2,3)
  24: (1,4)
  25: (1,3,1)

Links

Crossrefs

The opposite version is A374698, counted by A374687.
The weak version is A374701, counted by A374743.
For identical instead of distinct runs we have A374759, counted by A374760.
Compositions of this type are counted by A374761.
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.
- 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 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, A233564, A238343, A272919, A333213, A373949, A374633, A374685, A374744, A374758, A375128.

Programs

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

A374638 Numbers k such that the leaders of anti-runs in the k-th composition in standard order (A066099) are distinct.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 22, 24, 25, 26, 32, 33, 34, 35, 37, 38, 40, 41, 44, 45, 46, 48, 49, 50, 52, 53, 54, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 88, 89, 91, 92, 93, 96, 97, 98, 100, 101, 102, 104
Offset: 1

Views

Author

Gus Wiseman, Aug 01 2024

Keywords

Comments

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.

Examples

			The terms together with corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   6: (1,2)
   8: (4)
   9: (3,1)
  11: (2,1,1)
  12: (1,3)
  13: (1,2,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  20: (2,3)
  22: (2,1,2)
  24: (1,4)
  25: (1,3,1)
  26: (1,2,2)

Links

Crossrefs

Positions of distinct (strict) rows in A374515.
Compositions of this type are counted by A374518.
For identical instead of distinct we have A374519, counted by A374517.
The complement is A374639.
Other types of runs (instead of anti-):
- For identical runs we have A374249, counted by A274174.
- For weakly increasing runs we have A374768, counted by A374632.
- For strictly increasing runs we have A374698, counted by A374687.
- For weakly decreasing runs we have A374701, counted by A374743.
- For strictly decreasing runs we have A374767, counted by A374761.
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.
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, A335467, A373949.

Programs

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

A374519 Numbers k such that the leaders of anti-runs in the k-th composition in standard order (A066099) are identical.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 48, 49, 50, 51, 52, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 73, 76, 77, 80, 81, 82, 84, 85
Offset: 1

Views

Author

Gus Wiseman, Aug 01 2024

Keywords

Comments

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.

Examples

			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), so 346 is in the sequence.

Links

Crossrefs

Positions of constant rows in A374515.
Compositions of this type are counted by A374517.
The complement is A374520.
For distinct instead of identical leaders we have A374638, counted by A374518.
Other types of runs (instead of anti-):
- For identical runs we have A272919, counted by A000005.
- For weakly increasing runs we have A374633, counted by A374631.
- For strictly increasing runs we have A374685, counted by A374686.
- For weakly decreasing runs we have A374744, counted by A374742.
- For strictly decreasing runs we have A374759, counted by A374760.
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.
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, A373949.

Programs

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

A375123 Weakly increasing run-leader transformation for standard compositions.

Original entry on oeis.org

0, 1, 2, 1, 4, 5, 1, 1, 8, 9, 2, 5, 1, 3, 1, 1, 16, 17, 18, 9, 2, 5, 5, 5, 1, 3, 1, 3, 1, 3, 1, 1, 32, 33, 34, 17, 4, 37, 9, 9, 2, 5, 2, 5, 5, 11, 5, 5, 1, 3, 6, 3, 1, 3, 3, 3, 1, 3, 1, 3, 1, 3, 1, 1, 64, 65, 66, 33, 68, 69, 17, 17, 4, 9, 18, 37, 9, 19, 9, 9
Offset: 0

Views

Author

Gus Wiseman, Aug 02 2024

Keywords

Comments

The a(n)-th composition in standard order lists the leaders of weakly increasing runs of the n-th composition in standard order.
The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly 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.

Examples

			The 813th composition in standard order is (1,3,2,1,2,1), with weakly 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.

Links

Crossrefs

Positions of elements of A233564 are A374768, counted by A374632.
Positions of elements of A272919 are A374633, counted by A374631.
Ranks of rows of A374629.
The opposite version is A375124.
The strict version is A375125.
The strict opposite version is A375126.
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).
- Leader is A065120.
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627, sum A070939.
- Run-sum transformation is A353847.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
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. A189076, A238343, A333213, A373949, A374634, A374635, A374637, A374743, A374744, A375128.

Programs

  • Mathematica
    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],LessEqual]],{n,0,100}]

Formula

A000120(a(n)) = A124766(n).
A070939(a(n)) = A374630(n) for n > 0.
A065120(a(n)) = A065120(n).
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