A188920 -id:A188920 - cp's OEIS frontend

A374762 Number of integer compositions of n whose leaders of strictly decreasing runs are strictly increasing.

1, 1, 1, 3, 4, 6, 11, 18, 27, 41, 64, 98, 151, 229, 339, 504, 746, 1097, 1618, 2372, 3451, 5009, 7233, 10394, 14905, 21316, 30396, 43246, 61369, 86830, 122529, 172457, 242092, 339062, 473850, 660829, 919822, 1277935, 1772174, 2453151, 3389762, 4675660, 6438248

Offset: 0

Gus Wiseman, Jul 29 2024

nonn

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.

Also the number of ways to choose a strict integer partition of each part of an integer composition of n (A304969) such that the maxima are strictly decreasing. The weakly decreasing version is A374764.

			The a(0) = 1 through a(7) = 18 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)    (7)
                (12)  (13)   (14)   (15)   (16)
                (21)  (31)   (23)   (24)   (25)
                      (121)  (32)   (42)   (34)
                             (41)   (51)   (43)
                             (131)  (123)  (52)
                                    (132)  (61)
                                    (141)  (124)
                                    (213)  (142)
                                    (231)  (151)
                                    (321)  (214)
                                           (232)
                                           (241)
                                           (421)
                                           (1213)
                                           (1231)
                                           (1321)
                                           (2131)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For partitions instead of compositions we have A000009.
The weak version appears to be A188900.
The opposite version is A374689.
Other types of runs (instead of strictly decreasing):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we have A374634.
- For leaders of anti-runs we have A374679.
Other types of run-leaders (instead of strictly increasing):
- For identical leaders we have A374760, ranks A374759.
- For distinct leaders we have A374761, ranks A374767.
- For strictly decreasing leaders we have A374763.
- For weakly increasing leaders we have A374764.
- For weakly decreasing leaders we have A374765.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A106356, A188920, A189076, A238343, A261982, A333213, A374518, A374631, A374632, A374687, A374742, A374743.

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

seq(n) = Vec(prod(k=1, n, 1 + x^k*prod(j=1, min(n-k,k-1), 1 + x^j, 1 + O(x^(n-k+1))))) \\ Andrew Howroyd, Jul 31 2024

G.f.: Product_{k>=1} (1 + x^k*Product_{j=1..k-1} (1 + x^j)). - Andrew Howroyd, Jul 31 2024

a(24) onwards from Andrew Howroyd, Jul 31 2024

A374763 Number of integer compositions of n whose leaders of strictly decreasing runs are themselves strictly decreasing.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 10, 15, 22, 32, 47, 71, 106, 156, 227, 328, 473, 683, 986, 1421, 2040, 2916, 4149, 5882, 8314, 11727, 16515, 23221, 32593, 45655, 63810, 88979, 123789, 171838, 238055, 329187, 454451, 626412, 862164, 1184917, 1626124, 2228324, 3048982, 4165640, 5682847

Offset: 0

Gus Wiseman, Jul 30 2024

nonn

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 composition (3,1,2,1,1) has strictly decreasing runs ((3,1),(2,1),(1)), with leaders (3,2,1), so is counted under a(8).
The a(0) = 1 through a(8) = 15 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)    (7)     (8)
                (21)  (31)   (32)   (42)   (43)    (53)
                      (211)  (41)   (51)   (52)    (62)
                             (311)  (312)  (61)    (71)
                                    (321)  (322)   (413)
                                    (411)  (412)   (422)
                                           (421)   (431)
                                           (511)   (512)
                                           (3121)  (521)
                                           (3211)  (611)
                                                   (3212)
                                                   (3221)
                                                   (4121)
                                                   (4211)
                                                   (31211)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The opposite version is A374688.
The weak version is A374747.
For partitions instead of compositions we have A375133.
Other types of runs (instead of strictly decreasing):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we appear to have A188920.
- For leaders of anti-runs we have A374680.
- For leaders of strictly increasing runs we have A374689.
- For leaders of weakly decreasing runs we have A374746.
Other types of run-leaders (instead of strictly decreasing):
- For identical leaders we have A374760, ranks A374759.
- For distinct leaders we have A374761, ranks A374767.
- For strictly increasing leaders we have A374762.
- For weakly increasing leaders we have A374764.
- For weakly decreasing leaders we have A374765.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373949 counts compositions by run-compressed sum, opposite A373951.
Cf. A000009, A106356, A188900, A238343, A261982, A333213, A374518, A374632, A374635, A374687, A374742, A374743.

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

seq(n)={ my(A=O(x*x^n), p=1+A, q=p, r=p); for(k=1, n\2, r += x^k*q; p *= 1 + x^k; q *= 1 + x^k*p); Vec(r + x^(n\2+1)*q/(1-x)) } \\ Andrew Howroyd, Dec 30 2024

G.f.: Sum_{k>=0} x^k*Q(k,x) where Q(0,x) = 1 and Q(k,x) = Q(k-1,x) * (1 + x^k*Product_{j=1..k} (1 + x^j)) for k > 0. - Andrew Howroyd, Dec 30 2024

a(24) onwards from Andrew Howroyd, Dec 30 2024

A374764 Number of integer compositions of n whose leaders of strictly decreasing runs are weakly increasing.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 23, 40, 69, 118, 199, 333, 553, 911, 1492, 2428, 3928, 6323, 10129, 16151, 25646, 40560, 63905, 100332, 156995, 244877, 380803, 590479, 913100, 1408309, 2166671, 3325445, 5092283, 7780751, 11863546, 18052080, 27415291, 41556849, 62879053, 94975305, 143213145

Offset: 0

Gus Wiseman, Jul 30 2024

nonn

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.

Also the number of ways to choose a strict integer partition of each part of an integer composition of n (A304969) such that the maxima are weakly increasing [but weakly decreasing works too]. The strictly increasing version is A374762.

			The composition (1,1,2,1) has strictly decreasing runs ((1),(1),(2,1)) with leaders (1,1,2) so is counted under a(5).
The composition (1,2,1,1) has strictly decreasing runs ((1),(2,1),(1)) with leaders (1,2,1) so is not counted under a(5).
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)  (122)
                                (131)
                                (212)
                                (221)
                                (1112)
                                (1121)
                                (11111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For partitions instead of compositions we have A034296.
For strictly increasing leaders we have A374688.
The opposite version is A374697.
Other types of runs (instead of strictly decreasing):
- For leaders of identical runs we have A000041.
- For leaders of anti-runs we have A374681.
- For leaders of weakly increasing runs we have A374635.
- For leaders of strictly increasing runs we have A374690.
- For leaders of weakly decreasing runs we have A188900.
Other types of run-leaders (instead of weakly increasing):
- For identical leaders we have A374760, ranks A374759.
- For distinct leaders we have A374761, ranks A374767.
- For strictly increasing leaders we have A374762.
- For weakly decreasing leaders we have A374765.
- For strictly decreasing leaders we have A374763.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A335548 counts non-contiguous compositions, ranks A374253.
A373949 counts compositions by run-compressed sum, opposite A373951.
Cf. A106356, A188920, A238343, A261982, A333213, A374687, A374679, A374680, A374742, A374743, A374747.

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

seq(n) = Vec(1/prod(k=1, n, 1 - x^k*prod(j=1, min(n-k,k-1), 1 + x^j, 1 + O(x^(n-k+1))))) \\ Andrew Howroyd, Jul 31 2024

G.f.: 1/(Product_{k>=1} (1 - x^k*Product_{j=1..k-1} (1 + x^j))). - Andrew Howroyd, Jul 31 2024

a(24) onwards from Andrew Howroyd, Jul 31 2024

A375137 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 1-32.

Original entry on oeis.org

50, 98, 101, 114, 178, 194, 196, 197, 202, 203, 210, 226, 229, 242, 306, 324, 354, 357, 370, 386, 388, 389, 393, 394, 395, 402, 404, 405, 406, 407, 418, 421, 434, 450, 452, 453, 458, 459, 466, 482, 485, 498, 562, 610, 613, 626, 644, 649, 690, 706, 708, 709

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 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 (A375138) ranks compositions matching the dashed pattern 23-1.

			Composition 102 is (1,3,1,2), which matches 1-3-2 but not 1-32.
Composition 210 is (1,2,3,2), which matches 1-32 but not 132.
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 terms together with corresponding compositions begin:
   50: (1,3,2)
   98: (1,4,2)
  101: (1,3,2,1)
  114: (1,1,3,2)
  178: (2,1,3,2)
  194: (1,5,2)
  196: (1,4,3)
  197: (1,4,2,1)
  202: (1,3,2,2)
  203: (1,3,2,1,1)
  210: (1,2,3,2)
  226: (1,1,4,2)
  229: (1,1,3,2,1)
  242: (1,1,1,3,2)

Wikipedia, Permutation pattern.

The complement is too dense, but counted by A189076.
The non-dashed version is A335480, reverse A335482.
For leaders of identical runs we have A335485, reverse A335486.
For identical leaders we have A374633, counted by A374631.
Compositions of this type are counted by A374636.
For distinct leaders we have A374768, counted by A374632.
The reverse version is A375138, counted by A374636.
For leaders of strictly increasing runs we have A375139, counted by A375135.
Matching 1-21 also gives A375295, 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, A373948, A373953, A374634, A374635, A374637, A375123, A375296.

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

A188919 Triangle read by rows: T(n,k) = number of permutations of length n with k inversions that avoid the "dashed pattern" 1-32.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 3, 3, 1, 1, 1, 2, 4, 7, 8, 9, 9, 6, 4, 1, 1, 1, 2, 4, 7, 13, 16, 22, 26, 29, 26, 23, 17, 10, 5, 1, 1, 1, 2, 4, 7, 13, 22, 31, 44, 60, 74, 89, 95, 98, 93, 82, 63, 47, 29, 15, 6, 1, 1, 1, 2, 4, 7, 13, 22, 38, 55, 83, 116, 160, 207, 259, 304, 347, 375, 386, 378, 348, 304, 249, 190, 131, 85, 46, 21, 7, 1

Offset: 0

N. J. A. Sloane, Apr 13 2011

Row n has length 1 + binomial(n,2) and sum A000110(n) (a Bell number).

			Triangle begins:
1
1
1 1
1 1 2 1
1 1 2 4 3 3 1
1 1 2 4 7 8 9 9 6 4 1
...

Alois P. Heinz, Rows n = 0..50, flattened
A. M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011.
Andrew Baxter, Additional terms, formatted as a table.
Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv preprint arXiv:1108.2642, 2011
Jean-Christophe Novelli, Jean-Yves Thibon, Frédéric Toumazet, Noncommutative Bell polynomials and the dual immaculate basis, arXiv:1705.08113 [math.CO], 2017.

The column limits are given by A188920.

Cf. A000110, A161680.

b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
       add(b(u-j, o+j-1)*x^(o+j-1), j=1..u)+
       add(`if`(u=0, b(u+j-1, o-j)*x^(o-j), 0), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(0, n)):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 14 2015

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]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 01 2016, after Alois P. Heinz *)

More terms from Andrew Baxter, May 17 2011.

A374640 Number of integer compositions of n whose leaders of maximal anti-runs are not identical.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 7, 18, 43, 96, 211, 463, 992, 2112, 4462, 9347, 19495, 40480, 83690, 172478, 354455, 726538, 1486024, 3033644, 6182389, 12580486

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(7) = 18 compositions:
  .  .  .  .  (211)  (122)   (411)    (133)
                     (311)   (1122)   (322)
                     (2111)  (1221)   (511)
                             (2112)   (1222)
                             (2211)   (2113)
                             (3111)   (2311)
                             (21111)  (3112)
                                      (3211)
                                      (4111)
                                      (11122)
                                      (11221)
                                      (12211)
                                      (21112)
                                      (21121)
                                      (21211)
                                      (22111)
                                      (31111)
                                      (211111)

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

For partitions instead of compositions we have A239955.
The complement is counted by A374517, ranks A374519.
Compositions of this type are ranked by A374520, complement A374519.
For distinct instead of identical leaders we have A374678, ranks A374639, complement A374518, ranks A374638.
A003242 counts anti-runs, ranks A333489.
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A274174 counts contiguous compositions, ranks A374249.
Cf. A034296, A188920, A189076, A238343, A238424, A272919, A333213, A373949, A374515, A374700.

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

A374690 Number of integer compositions of n whose leaders of strictly increasing runs are weakly increasing.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 19, 34, 63, 115, 211, 387, 710, 1302, 2385, 4372, 8009, 14671, 26867, 49196, 90069, 164884, 301812, 552406, 1011004, 1850209, 3385861, 6195832, 11337470, 20745337, 37959030, 69454669, 127081111, 232517129, 425426211, 778376479, 1424137721

Offset: 0

Gus Wiseman, Jul 27 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 composition (1,1,3,2,3,2) has strictly increasing runs ((1),(1,3),(2,3),(2)), with leaders (1,1,2,2), so is counted under a(12).
The a(0) = 1 through a(6) = 19 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (12)   (13)    (14)     (15)
                 (111)  (22)    (23)     (24)
                        (112)   (113)    (33)
                        (121)   (122)    (114)
                        (1111)  (131)    (123)
                                (1112)   (132)
                                (1121)   (141)
                                (1211)   (222)
                                (11111)  (1113)
                                         (1122)
                                         (1131)
                                         (1212)
                                         (1311)
                                         (11112)
                                         (11121)
                                         (11211)
                                         (12111)
                                         (111111)

Christian Sievers, Table of n, a(n) for n = 0..500
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Ranked by positions of weakly increasing rows in A374683.
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have A000041.
- For leaders of anti-runs we have A374681.
- For leaders of weakly increasing runs we have A374635.
- For leaders of weakly decreasing runs we have A188900.
- For leaders of strictly decreasing runs we have A374764.
Types of run-leaders (instead of weakly increasing):
- For identical leaders we have A374686, ranks A374685.
- For distinct leaders we have A374687, ranks A374698.
- For strictly increasing leaders we have A374688.
- For strictly decreasing leaders we have A374689.
- For weakly decreasing leaders we have A374697.
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. A000009, A106356, A188920, A189076, A238343, A261982, A333213, A374629, A374630, A374632, A374679.

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

a(26) and beyond from Christian Sievers, Aug 08 2024

A374765 Number of integer compositions of n whose leaders of strictly decreasing runs are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 88, 141, 225, 357, 565, 891, 1399, 2191, 3420, 5321, 8256, 12774, 19711, 30339, 46584, 71359, 109066, 166340, 253163, 384539, 582972, 882166, 1332538, 2009377, 3024969, 4546562, 6822926, 10223632, 15297051, 22855872, 34103117

Offset: 0

Gus Wiseman, Jul 30 2024

nonn

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 composition (3,1,2,2,1) has strictly decreasing runs ((3,1),(2),(2,1)), with leaders (3,2,2), so is counted under a(9).
The a(0) = 1 through a(6) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (32)     (33)
                 (111)  (31)    (41)     (42)
                        (211)   (212)    (51)
                        (1111)  (221)    (222)
                                (311)    (312)
                                (2111)   (321)
                                (11111)  (411)
                                         (2121)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

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

The opposite version is A374690.
Other types of runs (instead of strictly decreasing):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we appear to have A189076.
- For leaders of anti-runs we have A374682.
- For leaders of strictly increasing runs we have A374697.
- For leaders of weakly decreasing runs we have A374747.
Other types of run-leaders (instead of weakly decreasing):
- For identical leaders we have A374760, ranks A374759.
- For distinct leaders we have A374761, ranks A374767.
- For strictly increasing leaders we have A374762.
- For strictly decreasing leaders we have A374763.
- For weakly increasing leaders we have A374764.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373949 counts compositions by run-compressed sum, opposite A373951.
Cf. A106356, A188900, A188920, A238343, A261982, A333213, A374635, A374636, A374689, A374742, A374743, A375133.

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

dfs(m, r, u) = 1 + sum(s=r, min(m, u), dfs(m-s, s, s)*x^s + sum(t=1, min(s-1, m-s), dfs(m-s-t, t, s)*x^(s+t)*prod(i=t+1, s-1, 1+x^i)));
lista(nn) = Vec(dfs(nn, 1, nn) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 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).

cp's OEIS Frontend

A374762 Number of integer compositions of n whose leaders of strictly decreasing runs are strictly increasing.

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A374763 Number of integer compositions of n whose leaders of strictly decreasing runs are themselves strictly decreasing.

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A374764 Number of integer compositions of n whose leaders of strictly decreasing runs are weakly increasing.

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A375137 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 1-32.

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A188919 Triangle read by rows: T(n,k) = number of permutations of length n with k inversions that avoid the "dashed pattern" 1-32.

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A374640 Number of integer compositions of n whose leaders of maximal anti-runs are not identical.

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A374690 Number of integer compositions of n whose leaders of strictly increasing runs are weakly increasing.

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