A375137 -id:A375137 - cp's OEIS frontend

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

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

Gus Wiseman, Jul 20 2024

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.

			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)

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

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.

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

A189076 Number of compositions of n that avoid the pattern 23-1.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 31, 61, 118, 228, 440, 846, 1623, 3111, 5955, 11385, 21752, 41530, 79250, 151161, 288224, 549408, 1047034, 1995000, 3800662, 7239710, 13789219, 26261678, 50012275, 95237360, 181350695, 345315255, 657506300, 1251912618, 2383636280, 4538364446

Offset: 0

N. J. A. Sloane, Apr 16 2011

nonn

Note that an exponentiation ^(-1) is missing in Example 4.4. The notation in Theorem 4.3 is complete.

Theorem: The reverse of a composition avoids 23-1 iff its leaders of maximal weakly increasing runs are weakly decreasing. For example, the composition y = (3,2,1,2,2,1,2,5,1,1,1) has maximal weakly increasing runs ((3),(2),(1,2,2),(1,2,5),(1,1,1)), with leaders (3,2,1,1,1), which are weakly decreasing, so the reverse of y is counted under a(21). - Gus Wiseman, Aug 19 2024

			From _Gus Wiseman_, Aug 19 2024: (Start)
The a(6) = 31 compositions:
  .  (6)  (5,1)  (4,1,1)  (3,1,1,1)  (2,1,1,1,1)  (1,1,1,1,1,1)
          (1,5)  (1,4,1)  (1,3,1,1)  (1,2,1,1,1)
          (4,2)  (1,1,4)  (1,1,3,1)  (1,1,2,1,1)
          (2,4)  (3,2,1)  (1,1,1,3)  (1,1,1,2,1)
          (3,3)  (3,1,2)  (2,2,1,1)  (1,1,1,1,2)
                 (2,3,1)  (2,1,2,1)
                 (2,1,3)  (2,1,1,2)
                 (1,2,3)  (1,2,2,1)
                 (2,2,2)  (1,2,1,2)
                          (1,1,2,2)
Missing is (1,3,2), reverse of (2,3,1).
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..300
S. Heubach, T. Mansour and A. O. Munagi, Avoiding Permutation Patterns of Type (2,1) in Compositions, Online Journal of Analytic Combinatorics, 4 (2009).
Wikipedia, Permutation pattern.
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The non-dashed version is A102726.
The version for 3-12 is A188900, complement A375406.
Avoiding 12-1 also gives A188920 in reverse.
The version for 13-2 is A189077.
For identical leaders we have A374631, ranks A374633.
For distinct leaders we have A374632, ranks A374768.
The complement is counted by A374636, ranks A375137.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
Cf. A056823, A188919, A238343, A274174, A333213, A335480, A358836, A374635.

A189075 := proc(n) local g,i; g := 1; for i from 1 to n do 1-x^i/mul ( 1-x^j,j=i+1..n-i) ; g := g*% ; end do: g := expand(1/g) ; g := taylor(g,x=0,n+1) ; coeftayl(g,x=0,n) ; end proc: # R. J. Mathar, Apr 16 2011

a[n_] := Module[{g = 1, xi}, Do[xi = 1 - x^i/Product[1 - x^j, {j, i+1, n-i}]; g = g xi, {i, n}]; SeriesCoefficient[1/g, {x, 0, n}]];
a /@ Range[0, 32] (* Jean-François Alcover, Apr 02 2020, after R. J. Mathar *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,y_,z_,_,x_,_}/;xGus Wiseman, Aug 19 2024 *)

A188920 a(n) is the limiting term of the n-th column of the triangle in A188919.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 22, 38, 63, 105, 169, 274, 434, 686, 1069, 1660, 2548, 3897, 5906, 8911, 13352, 19917, 29532, 43605, 64056, 93715, 136499, 198059, 286233, 412199, 591455, 845851, 1205687, 1713286, 2427177, 3428611, 4829563, 6784550, 9505840, 13284849

Offset: 0

N. J. A. Sloane, Apr 13 2011

nonn

Also the number of integer compositions of n whose reverse avoids 12-1 and 23-1.

Theorem: The reverse of a composition avoids 12-1 and 23-1 iff its leaders of maximal weakly increasing runs, obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each, are strictly decreasing. For example, the composition y = (4,5,3,2,2,3,1,3,5) has reverse (5,3,1,3,2,2,3,5,4), which avoids 12-1 and 23-1, while the maximal weakly increasing runs of y are ((4,5),(3),(2,2,3),(1,3,5)), with leaders (4,3,2,1), which are strictly decreasing, as required. - Gus Wiseman, Aug 20 2024

			From _Gus Wiseman_, Aug 20 2024: (Start)
The a(0) = 1 through a(6) = 22 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (12)   (13)    (14)     (15)
                 (21)   (22)    (23)     (24)
                 (111)  (31)    (32)     (33)
                        (112)   (41)     (42)
                        (211)   (113)    (51)
                        (1111)  (122)    (114)
                                (212)    (123)
                                (221)    (132)
                                (311)    (213)
                                (1112)   (222)
                                (2111)   (312)
                                (11111)  (321)
                                         (411)
                                         (1113)
                                         (1122)
                                         (2112)
                                         (2211)
                                         (3111)
                                         (11112)
                                         (21111)
                                         (111111)
(End)

John Tyler Rascoe, Table of n, a(n) for n = 0..200
A. M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011.
Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv preprint arXiv:1108.2642 [math.CO], 2011-2012.
Wikipedia, Permutation pattern.
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For leaders of identical runs we have A000041.
Matching 23-1 only gives A189076.
An opposite version is A358836.
For identical leaders we have A374631, ranks A374633.
For distinct leaders we have A374632, ranks A374768.
For weakly increasing leaders we have A374635.
For non-weakly decreasing leaders we have A374636, ranks A375137.
For leaders of anti-runs we have A374680.
For leaders of strictly increasing runs we have A374689.
The complement is counted by A375140, ranks A375295, reverse A375296.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
Cf. A056823, A102726, A188900, A188919, A189077, A238343, A333213, A335480/A335482, A374679, A374688.

b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1, Sum[b[u - j, o + j - 1]*x^(o + j - 1), {j, 1, u}] + Sum[If[u == 0, b[u + j - 1, o - j]*x^(o - j), 0], {j, 1, o}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[0, n]];
Take[T[40], 40] (* Jean-François Alcover, Sep 15 2018, after Alois P. Heinz in A188919 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Greater@@First/@Split[Reverse[#],LessEqual]&]],{n,0,15}] (* Gus Wiseman, Aug 20 2024 *)
- or -
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !MatchQ[#,{_,y_,z_,_,x_,_}/;x<=yGus Wiseman, Aug 20 2024 *)

B_x(i,N) = {my(x='x+O('x^N), f=(x^i)/(1-x^i)*prod(j=i+1,N-i,1/(1-x^j))); f}
A_x(N) = {my(x='x+O('x^N), f=1+sum(i=1,N, B_x(i,N)*prod(j=1,i-1,1+B_x(j,N)))); Vec(f)}
A_x(60) \\ John Tyler Rascoe, Aug 23 2024

a(n) = 2^(n-1) - A375140(n).

G.f.: 1 + Sum_{i>0} (B(i,x) * Product_{j=1..i-1} (1 + B(j,x))) where B(i,x) = (x^i)/(1-x^i) * Product_{j>i} (1/(1-x^j)). - John Tyler Rascoe, Aug 23 2024

More terms from Andrew Baxter, May 17 2011

a(30)-a(39) from Alois P. Heinz, Nov 14 2015

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

A375138 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 23-1.

Original entry on oeis.org

41, 81, 83, 105, 145, 161, 163, 165, 166, 167, 169, 209, 211, 233, 289, 290, 291, 297, 321, 323, 325, 326, 327, 329, 331, 332, 333, 334, 335, 337, 339, 361, 401, 417, 419, 421, 422, 423, 425, 465, 467, 489, 545, 553, 577, 578, 579, 581, 582, 583, 593, 595, 617

Offset: 1

Gus Wiseman, Aug 09 2024

nonn

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.
These are also numbers k such that the maximal weakly increasing runs in the reverse of the k-th composition in standard order do not have weakly decreasing leaders, where 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 reverse version (A375137) ranks compositions matching the dashed pattern 1-32.

			Composition 89 is (2,1,3,1), which matches 2-3-1 but not 23-1.
Composition 165 is (2,3,2,1), which matches 23-1 but not 231.
Composition 358 is (2,1,3,1,2), which matches 2-3-1 and 1-3-2 but not 23-1 or 1-32.
The sequence together with corresponding compositions begins:
   41: (2,3,1)
   81: (2,4,1)
   83: (2,3,1,1)
  105: (1,2,3,1)
  145: (3,4,1)
  161: (2,5,1)
  163: (2,4,1,1)
  165: (2,3,2,1)
  166: (2,3,1,2)
  167: (2,3,1,1,1)
  169: (2,2,3,1)
  209: (1,2,4,1)
  211: (1,2,3,1,1)
  233: (1,1,2,3,1)

Wikipedia, Permutation pattern.

The complement is too dense, but counted by A189076.
The non-dashed version is A335482, reverse A335480.
For leaders of identical runs we have A335486, reverse A335485.
Compositions of this type are counted by A374636.
The reverse version is A375137, counted by A374636.
Matching 12-1 also gives A375296, counted by A375140 (complement A188920).
A003242 counts anti-runs, ranks A333489.
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, reverse A228351.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Run-length transform is A333627, sum A070939.
- Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
Cf. A056823, A106356, A188919, A238343, A333213, A335466, A373948, A373953, A374633, A375123, A375139, A374768.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MatchQ[stc[#],{_,y_,z_,_,x_,_}/;x

A375140 Number of integer compositions of n whose leaders of weakly increasing runs are not strictly decreasing.

Original entry on oeis.org

0, 0, 0, 1, 3, 10, 26, 65, 151, 343, 750, 1614, 3410, 7123, 14724, 30220, 61639, 125166, 253233, 510936, 1028659, 2067620, 4150699, 8324552, 16683501, 33417933, 66910805, 133931495, 268023257, 536279457, 1072895973, 2146277961, 4293254010, 8587507415

Offset: 1

Gus Wiseman, Aug 10 2024

nonn

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.

Also the number of integer compositions of n matching the dashed patterns 1-32 or 1-21.

			The a(1) = 0 through a(6) = 10 compositions:
     .  .  .  (121)  (131)   (132)
                     (1121)  (141)
                     (1211)  (1131)
                             (1212)
                             (1221)
                             (1311)
                             (2121)
                             (11121)
                             (11211)
                             (12111)

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

For leaders of identical runs we have A056823.
The complement is counted by A188920.
Leaders of weakly increasing runs are rows of A374629, sum A374630.
For weakly decreasing leaders we have A374636, ranks A375137 or A375138.
For leaders of weakly decreasing runs we have the complement of A374746.
Compositions of this type are ranked by A375295, reverse A375296.
A003242 counts anti-runs, ranks A333489.
A106356 counts compositions by number of maximal anti-runs.
A238424 counts partitions whose first differences are an anti-run.
A274174 counts contiguous compositions, ranks A374249.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
A335548 counts non-contiguous compositions, ranks A374253.
A374637 counts compositions by sum of leaders of weakly increasing runs.
Cf. A124766, A238343, A261982, A333213, A335514, A373949, A374635, A374678, A374687, A374761.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],!Greater@@First/@Split[#,LessEqual]&]],{n,15}]
- or -
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{_,x_,_,z_,y_,_}/;x<=y

a(n) = 2^(n-1) - A188920(n).

A375295 Numbers k such that the leaders of maximal weakly increasing runs in the k-th composition in standard order (row k of A066099) are not strictly decreasing.

Original entry on oeis.org

13, 25, 27, 29, 45, 49, 50, 51, 53, 54, 55, 57, 59, 61, 77, 82, 89, 91, 93, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 123, 125, 141, 153, 155, 157, 162, 165, 173, 177, 178, 179, 181, 182, 183, 185, 187, 189

Offset: 1

Gus Wiseman, Aug 12 2024

nonn

First differs from the non-dashed version in lacking 166, corresponding to the composition (2,3,1,2).
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.
Also numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed patterns 1-32 or 1-21.

			The sequence together with corresponding compositions begins:
  13: (1,2,1)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  45: (2,1,2,1)
  49: (1,4,1)
  50: (1,3,2)
  51: (1,3,1,1)
  53: (1,2,2,1)
  54: (1,2,1,2)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)
  61: (1,1,1,2,1)
  77: (3,1,2,1)
  82: (2,3,2)
  89: (2,1,3,1)
  91: (2,1,2,1,1)
  93: (2,1,1,2,1)

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

For leaders of identical runs we have A335485.
Positions of non-strictly decreasing rows in A374629 (sums A374630).
For identical leaders we have A374633, counted by A374631.
Matching 1-32 only gives A375137, reverse A375138, both counted by A374636.
Interchanging weak/strict gives A375139, counted by A375135.
Compositions of this type are counted by A375140, complement A188920.
The reverse version is A375296.
A003242 counts anti-runs, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A374637 counts compositions by sum of leaders of weakly 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, reverse A228351.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Run-length transform is A333627, sum A070939.
- Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
Cf. A056823, A106356, A188919, A189076, A238343, A261982, A333213, A335480, A335482, A373948, A374746, A374768, A375123.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!Greater@@First/@Split[stc[#],LessEqual]&]
- or -
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,z_,y_,_}/;x<=y

A375296 Numbers k such that the leaders of maximal weakly increasing runs in the reverse of the k-th composition in standard order (row k of A228351) are not strictly decreasing.

Original entry on oeis.org

13, 25, 27, 29, 41, 45, 49, 51, 53, 54, 55, 57, 59, 61, 77, 81, 82, 83, 89, 91, 93, 97, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 115, 117, 118, 119, 121, 123, 125, 141, 145, 153, 155, 157, 161, 162, 163, 165, 166, 167, 169, 173, 177, 179, 181, 182

Offset: 1

Gus Wiseman, Aug 13 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.
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.
Also numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed patterns 23-1 or 12-1.

			The sequence together with corresponding compositions begins:
  13: (1,2,1)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  41: (2,3,1)
  45: (2,1,2,1)
  49: (1,4,1)
  51: (1,3,1,1)
  53: (1,2,2,1)
  54: (1,2,1,2)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)
  61: (1,1,1,2,1)

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

For leaders of identical runs we have A335486, reverse A335485.
Matching 1-32 only gives A375138, reverse A375137, both counted by A374636.
Compositions of this type are counted by A375140, complement A188920.
The reverse version is A375295.
A003242 counts anti-runs, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A374637 counts compositions by sum of leaders of weakly 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, reverse A228351.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Run-length transform is A333627, sum A070939.
- Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
Cf. A056823, A106356, A188919, A189076, A238343, A333213, A335480, A335482, A373948, A374630, A374633, A374768, A375123.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!Greater@@First/@Split[Reverse[stc[#]],LessEqual]&]
- or -
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,300],MatchQ[stc[#],{_,y_,z_,_,x_,_}/;x<=y

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]&]

A375297 Number of integer compositions of n matching both of the dashed patterns 23-1 and 1-32.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 21, 68, 199, 545, 1410, 3530, 8557, 20255, 46968, 107135, 240927, 535379, 1177435, 2566618, 5551456

Offset: 0

Gus Wiseman, Aug 23 2024

Also the number of integer compositions of n whose leaders of maximal weakly increasing runs are not weakly decreasing and whose reverse satisfies the same condition.

			The a(0) = 0 through a(11) = 21 compositions:
  .  .  .  .  .  .  .  .  .  (12321)  (1342)    (1352)
                                      (2431)    (2531)
                                      (12421)   (11342)
                                      (13231)   (12431)
                                      (112321)  (12521)
                                      (123211)  (13241)
                                                (13421)
                                                (14231)
                                                (23132)
                                                (24311)
                                                (112421)
                                                (113231)
                                                (122321)
                                                (123212)
                                                (123221)
                                                (124211)
                                                (132311)
                                                (212321)
                                                (1112321)
                                                (1123211)
                                                (1232111)

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

For leaders of identical runs we have A332834.
For just one of the two conditions we have A374636, ranks A375137/A375138.
These compositions are ranked by A375407.
A003242 counts anti-runs, ranks A333489.
A011782 counts compositions.
A106356 counts compositions by number of maximal anti-runs.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A335456 counts patterns matched by compositions.
Cf. A000041, A056823, A188920, A189076, A238343, A333213, A335514, A374631, A374632, A374635, A374681.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MatchQ[#,{_,y_,z_,_,x_,_}/;x_,x_,_,z_,y_,_}/;x

cp's OEIS Frontend

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

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A189076 Number of compositions of n that avoid the pattern 23-1.

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A188920 a(n) is the limiting term of the n-th column of the triangle in A188919.

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

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A375138 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 23-1.

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A375140 Number of integer compositions of n whose leaders of weakly increasing runs are not strictly decreasing.

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A375295 Numbers k such that the leaders of maximal weakly increasing runs in the k-th composition in standard order (row k of A066099) are not strictly decreasing.

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A375296 Numbers k such that the leaders of maximal weakly increasing runs in the reverse of the k-th composition in standard order (row k of A228351) are not strictly decreasing.

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