A374699 -id:A374699 - cp's OEIS frontend

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

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

A374682 Number of integer compositions of n whose leaders of anti-runs are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 30, 59, 114, 222, 434, 844, 1641, 3189, 6192, 12020, 23320, 45213, 87624, 169744, 328684, 636221, 1231067, 2381269, 4604713, 8901664

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(5) = 15 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (131)
                        (1111)  (212)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)

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

For reversed partitions instead of compositions we have A115029.
The complement is A374699.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we have A189076, complement A374636.
- For leaders of weakly decreasing runs we have A374747.
- For leaders of strictly decreasing runs we have A374765.
- For leaders of strictly increasing runs we have A374697.
Other types of run-leaders (instead of weakly decreasing):
- For identical leaders we have A374517, ranks A374519.
- For distinct leaders we have A374518, ranks A374638.
- For weakly increasing leaders we have A374681.
- For strictly increasing leaders we have A374679.
- 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. A238343, A333213, A333381, A373949, A374515, A374632, A374635, A374678, A374700, A374706.

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

A375135 Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 9, 25, 63, 152, 355, 809, 1804, 3963, 8590, 18423, 39161, 82620, 173198, 361101, 749326, 1548609, 3189132, 6547190, 13404613, 27378579, 55801506, 113517749, 230544752, 467519136, 946815630, 1915199736, 3869892105, 7812086380, 15756526347

Offset: 0

Gus Wiseman, Aug 06 2024

nonn

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

			The composition y = (1,2,1,3,2,3) has strictly increasing runs ((1,2),(1,3),(2,3)), with leaders (1,1,2), which are not weakly decreasing, so y is counted under a(12).
The a(0) = 0 through a(8) = 25 compositions:
  .  .  .  .  .  (122)  (132)   (133)    (143)
                        (1122)  (142)    (152)
                        (1221)  (1132)   (233)
                                (1222)   (1133)
                                (1321)   (1142)
                                (2122)   (1223)
                                (11122)  (1232)
                                (11221)  (1322)
                                (12211)  (1331)
                                         (1421)
                                         (2132)
                                         (3122)
                                         (11132)
                                         (11222)
                                         (11321)
                                         (12122)
                                         (12212)
                                         (12221)
                                         (13211)
                                         (21122)
                                         (21221)
                                         (111122)
                                         (111221)
                                         (112211)
                                         (122111)

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

For leaders of constant runs we have A056823.
For leaders of weakly increasing runs we have A374636, complement A189076?
The complement is counted by A374697.
For leaders of anti-runs we have A374699, complement A374682.
Other functional neighbors: A188920, A374764, A374765.
A003242 counts anti-run compositions, ranks A333489.
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.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A106356, A238343, A261982, A333213, A374632, A374679, A374683, A374689.

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

a(n) = A011782(n) - A374697(n). - Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 12 2024

nonn

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

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

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

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

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

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

cp's OEIS Frontend

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

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A374682 Number of integer compositions of n whose leaders of anti-runs are weakly decreasing.

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A375135 Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.

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

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Mathematica