A374638 -id:A374638 - cp's OEIS frontend

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

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}]

A374521 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of anti-runs sum to k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 1, 1, 2, 0, 2, 1, 2, 3, 0, 2, 5, 3, 4, 2, 0, 5, 7, 8, 3, 5, 4, 0, 9, 12, 11, 17, 5, 8, 2, 0, 14, 26, 23, 22, 24, 6, 9, 4, 0, 25, 42, 54, 41, 36, 36, 7, 12, 3, 0, 46, 76, 88, 107, 60, 60, 48, 9, 14, 4

Offset: 0

Gus Wiseman, Aug 02 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.

			Triangle begins:
   1
   0   1
   0   0   2
   0   1   1   2
   0   2   1   2   3
   0   2   5   3   4   2
   0   5   7   8   3   5   4
   0   9  12  11  17   5   8   2
   0  14  26  23  22  24   6   9   4
   0  25  42  54  41  36  36   7  12   3
   0  46  76  88 107  60  60  48   9  14   4
   0  78 144 166 179 176 101  83  68  10  17   2
   0 136 258 327 339 311 299 139 122  81  12  18   6
   0 242 457 602 704 591 544 447 198 165 109  12  23   2
Row n = 6 counts the following compositions:
  .  (15)    (24)    (321)    (42)     (51)     (6)
     (141)   (114)   (312)    (1122)   (411)    (33)
     (132)   (231)   (1113)   (11112)  (3111)   (222)
     (123)   (213)   (2112)            (2211)   (111111)
     (1212)  (1311)  (1221)            (21111)
             (1131)  (12111)
             (2121)  (11211)
                     (11121)

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

Column n = k is A000005, except a(0) = 1.
Row-sums are A011782.
Column k = 1 is A096569.
For length instead of sum we have A106356.
The corresponding rank statistic is A374516, row-sums of A374515.
For identical leaders we have A374517, ranks A374519.
For distinct leaders we have A374518, ranks A374638.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A373949.
- For leaders of weakly increasing runs we have A374637.
- For leaders of strictly increasing runs we have A374700.
- For leaders of weakly decreasing runs we have A374748.
- For leaders of strictly decreasing runs we have A374766.
A003242 counts anti-run compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
Cf. A124766, A238343, A261982, A333213, A374251, A374687, A374761.

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

A374699 Number of integer compositions of n whose leaders of maximal anti-runs are not weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 5, 14, 34, 78, 180, 407, 907, 2000, 4364, 9448, 20323, 43448, 92400, 195604, 412355, 866085, 1813035, 3783895, 7875552

Offset: 0

Gus Wiseman, Aug 06 2024

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 a(0) = 0 through a(8) = 14 compositions:
  .  .  .  .  .  (122)  (1122)  (133)    (233)
                        (1221)  (1222)   (1133)
                                (11122)  (1223)
                                (11221)  (1322)
                                (12211)  (1331)
                                         (11222)
                                         (12122)
                                         (12212)
                                         (12221)
                                         (21122)
                                         (111122)
                                         (111221)
                                         (112211)
                                         (122111)

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

The complement is counted by A374682.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A056823.
- For leaders of weakly increasing runs we have A374636, complement A189076?
- For leaders of strictly increasing runs: A375135, complement A374697.
Other types of run-leaders (instead of weakly decreasing):
- For identical leaders we have A374640, ranks A374520, complement A374517, ranks A374519.
- For distinct leaders we have A374678, ranks A374639, complement A374518, ranks A374638.
- For weakly increasing leaders we have complement A374681.
- For strictly increasing leaders we have complement complement A374679.
- For strictly decreasing leaders we have complement 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.
A333381 counts maximal anti-runs in standard compositions.
Cf. A115029, A238343, A333213, A373949, A374515, A374632, A374635, A374700.

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A374521 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of anti-runs sum to k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A374699 Number of integer compositions of n whose leaders of maximal anti-runs are not weakly decreasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica