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 15 results. Next

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)
		

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.

Programs

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

A374630 Sum of leaders of weakly increasing runs in the n-th composition in standard order.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jul 20 2024

Keywords

Comments

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 maximal weakly increasing subsequences of the 1234567th composition in standard order are ((3),(2),(1,2,2),(1,2,5),(1,1,1)), so a(1234567) = 8.
		

Crossrefs

For length instead of sum we have A124766.
For leaders of constant runs we have A373953, excess A373954.
For leaders of anti-runs we have A374516.
Row-sums of A374629.
Counting compositions by this statistic gives A374637.
For leaders of strictly increasing runs we have A374684.
For leaders of weakly decreasing runs we have A374741.
For leaders of strictly decreasing runs we have A374758
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
A373949 counts compositions by run-compressed sum, opposite A373951.
All of the following pertain to compositions in standard order:
- Ones are counted by A000120.
- Sum is A029837 (or sometimes A070939).
- Listed by A066099.
- Length is A070939.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of strict compositions are A233564, counted by A032020.
- Constant compositions are ranked by A272919.
- Ranks of anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627.
- Run-compression transform is A373948.

Programs

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

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)
		

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:

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)
		

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:

Programs

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

A374706 Sum of minima of the maximal strictly increasing runs in the weakly increasing prime indices of n.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Aug 04 2024

Keywords

Comments

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.

Examples

			The prime indices of 540 are {1,1,2,2,2,3}, with strictly increasing runs ({1},{1,2},{2},{2,3}), with minima (1,1,2,2), summing to a(540) = 6.
		

Crossrefs

For leaders of constant runs we have A066328.
A version for compositions is A374684, row-sums of A374683 (length A124768).
Row-sums of A375128.
For length instead of sum we have A375136.
A055887 counts sequences of partitions with total sum n.
A112798 lists prime indices:
- length A001222, distinct A001221
- leader A055396
- sum A056239
- reverse A296150

Programs

  • Mathematica
    prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
    Table[Total[First/@Split[prix[n],Less]],{n,100}]

A374759 Numbers k such that the leaders of strictly decreasing runs in the k-th composition in standard order are identical.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 15, 16, 17, 18, 21, 22, 31, 32, 33, 34, 36, 37, 42, 45, 63, 64, 65, 66, 68, 69, 73, 76, 85, 86, 90, 127, 128, 129, 130, 132, 133, 136, 137, 146, 148, 153, 170, 173, 181, 182
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.

Examples

			The 18789th composition in standard order is (3,3,2,1,3,2,1), with strictly decreasing runs ((3),(3,2,1),(3,2,1)), with leaders (3,3,3), so 18789 is in the sequence.
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)
  15: (1,1,1,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  21: (2,2,1)
  22: (2,1,2)
  31: (1,1,1,1,1)
  32: (6)
  33: (5,1)
  34: (4,2)
  36: (3,3)
  37: (3,2,1)
		

Crossrefs

For leaders of anti-runs we have A374519 (counted by A374517).
For leaders of weakly increasing runs we have A374633, counted by A374631.
The opposite version is A374685 (counted by A374686).
The weak version is A374744.
Compositions of this type are counted by A374760.
For distinct instead of identical runs we have 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:

Programs

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

A375133 Number of integer partitions of n whose maximal anti-runs have distinct maxima.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 8, 10, 14, 17, 23, 29, 38, 47, 60, 74, 93, 113, 141, 171, 211, 253, 309, 370, 447, 532, 639, 758, 904, 1066, 1265, 1487, 1754, 2053, 2411, 2813, 3289, 3823, 4454, 5161, 5990, 6920, 8005, 9223, 10634, 12218, 14048, 16101, 18462, 21107
Offset: 0

Views

Author

Gus Wiseman, Aug 14 2024

Keywords

Comments

An anti-run is a sequence with no adjacent equal parts.
These are partitions with no part appearing more than twice and greatest part appearing only once.
Also the number of reversed integer partitions of n whose maximal anti-runs have distinct maxima.

Examples

			The partition y = (6,5,5,4,3,3,2,1) has maximal anti-runs ((6,5),(5,4,3),(3,2,1)), with maxima (6,5,3), so y is counted under a(29).
The a(0) = 1 through a(9) = 14 partitions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)    (7)     (8)     (9)
                (21)  (31)   (32)   (42)   (43)    (53)    (54)
                      (211)  (41)   (51)   (52)    (62)    (63)
                             (311)  (321)  (61)    (71)    (72)
                                    (411)  (322)   (422)   (81)
                                           (421)   (431)   (432)
                                           (511)   (521)   (522)
                                           (3211)  (611)   (531)
                                                   (3221)  (621)
                                                   (4211)  (711)
                                                           (4221)
                                                           (4311)
                                                           (5211)
                                                           (32211)
		

Crossrefs

Includes all strict partitions A000009.
For identical instead of distinct see: A034296, A115029, A374760, A374759.
For compositions instead of partitions we have A374761.
For minima instead of maxima we have A375134, ranks A375398.
The complement is counted by A375401, ranks A375403.
These partitions are ranked by A375402, for compositions A374767.
The complement for minima instead of maxima is A375404, ranks A375399.
A000041 counts integer partitions.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts integer compositions.
A055887 counts sequences of partitions with total sum n.
A375128 lists minima of maximal anti-runs of prime indices, sums A374706.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n], UnsameQ@@Max/@Split[#,UnsameQ]&]],{n,0,30}]
  • PARI
    A_x(N) = {my(x='x+O('x^N), f=sum(i=0,N,(x^i)*prod(j=1,i-1,(1-x^(3*j))/(1-x^j)))); Vec(f)}
    A_x(51) \\ John Tyler Rascoe, Aug 21 2024

Formula

G.f.: Sum_{i>=0} (x^i * Product_{j=1..i-1} (1-x^(3*j))/(1-x^j)). - John Tyler Rascoe, Aug 21 2024

A375128 Irregular triangle read by rows where row n lists the minima of maximal strictly increasing runs in the weakly increasing prime indices of n.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 4, 1, 1, 1, 2, 2, 1, 5, 1, 1, 6, 1, 2, 1, 1, 1, 1, 7, 1, 2, 8, 1, 1, 2, 1, 9, 1, 1, 1, 3, 3, 1, 2, 2, 2, 1, 1, 10, 1, 11, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 2, 12, 1, 2, 1, 1, 1, 13, 1, 14, 1, 1, 2, 2, 1, 15, 1, 1, 1, 1, 4, 4, 1, 3, 2, 1, 1, 16
Offset: 1

Views

Author

Gus Wiseman, Aug 04 2024

Keywords

Comments

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

Examples

			The prime indices of 540 are {1,1,2,2,2,3}, with strictly increasing runs ({1},{1,2},{2},{2,3}), with minima (1,1,2,2), which is row 540.
Triangle begins:
   1:
   2:  1
   3:  2
   4:  1  1
   5:  3
   6:  1
   7:  4
   8:  1  1  1
   9:  2  2
  10:  1
  11:  5
  12:  1  1
  13:  6
  14:  1
  15:  2
  16:  1  1  1  1
		

Crossrefs

Row-minima are A055396.
Row-sums are A374706.
Row-lengths are A375136.
For leaders of constant runs we have A304038, row-sums A066328.
For compositions we have A374683, row-sums of A374684 (length A124768).
A112798 lists prime indices:
- length A001222, distinct A001221
- leader A055396
- sum A056239
- reverse A296150

Programs

  • Mathematica
    Table[If[n==1,{},First/@Split[Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]],Less]],{n,100}]

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

Views

Author

Gus Wiseman, Aug 02 2024

Keywords

Comments

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.

Examples

			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.
		

Crossrefs

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:

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],GreaterEqual]],{n,0,100}]

Formula

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

Views

Author

Gus Wiseman, Aug 02 2024

Keywords

Comments

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?

Examples

			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.
		

Crossrefs

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:

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],Greater]],{n,0,100}]

Formula

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