A261982 -id:A261982 - cp's OEIS frontend

A363262 Number of integer compositions of n in which the greatest part appears more than once.

0, 1, 1, 2, 4, 9, 18, 37, 73, 145, 287, 570, 1134, 2264, 4526, 9061, 18152, 36374, 72884, 146011, 292416, 585422, 1171632, 2344136, 4688821, 9376832, 18749169, 37485358, 74939850, 149813328, 299492966, 598729533, 1196987066, 2393137399, 4784846896, 9567357951

Offset: 1

Gus Wiseman, Jun 04 2023

nonn

Also the number of multisets of length n covering an initial interval of positive integers with more than one mode.

			The a(2) = 1 through a(6) = 9 compositions:
  (11)  (111)  (22)    (122)    (33)
               (1111)  (212)    (222)
                       (221)    (1122)
                       (11111)  (1212)
                                (1221)
                                (2112)
                                (2121)
                                (2211)
                                (111111)

For partitions instead of compositions we have A002865.
The complement is counted by A097979 shifted left.
Row sums of columns k > 1 of A238341.
If all parts appear more than once we have A240085, for partitions A007690.
If the greatest part appears exactly twice we have A243737.
For least instead of greatest we have A363224, see triangle A238342.
A000041 counts integer partitions, strict A000009.
A032020 counts strict compositions.
A067029 gives last exponent in prime factorization, first A071178.
A261982 counts compositions with some part appearing more than once.
Cf. A008284, A105039, A117989, A362607, A362608, A362612, A362614.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Count[#,Max@@#]>1&]],{n,15}]

A374766 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of maximal strictly decreasing runs sum to k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 3, 5, 0, 0, 0, 1, 8, 7, 0, 0, 0, 1, 3, 17, 11, 0, 0, 0, 0, 4, 10, 35, 15, 0, 0, 0, 0, 1, 12, 28, 65, 22, 0, 0, 0, 0, 1, 6, 31, 70, 118, 30, 0, 0, 0, 0, 1, 3, 22, 78, 163, 203, 42, 0, 0, 0, 0, 0, 4, 13, 69, 186, 354, 342, 56

Offset: 0

Gus Wiseman, Aug 02 2024

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.

Are the column-sums finite?

			Triangle begins:
   1
   0   1
   0   0   2
   0   0   1   3
   0   0   0   3   5
   0   0   0   1   8   7
   0   0   0   1   3  17  11
   0   0   0   0   4  10  35  15
   0   0   0   0   1  12  28  65  22
   0   0   0   0   1   6  31  70 118  30
   0   0   0   0   1   3  22  78 163 203  42
   0   0   0   0   0   4  13  69 186 354 342  56
Row n = 6 counts the following compositions:
  .  .  .  (321)  (42)    (51)     (6)
                  (132)   (411)    (15)
                  (2121)  (141)    (24)
                          (312)    (114)
                          (231)    (33)
                          (213)    (123)
                          (3111)   (1113)
                          (1311)   (222)
                          (1131)   (1122)
                          (2211)   (11112)
                          (2112)   (111111)
                          (1221)
                          (1212)
                          (21111)
                          (12111)
                          (11211)
                          (11121)

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

Column n = k is A000041.
Row-sums are A011782.
For length instead of sum we have A333213.
The corresponding rank statistic is A374758, row-sums of A374757.
For identical leaders we have A374760, ranks A374759.
For distinct leaders we have A374761, ranks A374767.
Other types of runs (instead of strictly decreasing):
- For leaders of identical runs we have A373949.
- For leaders of anti-runs we have A374521.
- 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.
A003242 counts anti-run compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
Cf. A106356, A238343, A261982, A274174, A374517, A374518, A374687.

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

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

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

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]&]

A262047 Number of ordered partitions of [n] such that at least two parts have the same size.

Original entry on oeis.org

0, 0, 2, 6, 66, 510, 4280, 46536, 542962, 7074654, 101914512, 1621871196, 28087868160, 526841965260, 10641234260358, 230278335503586, 5315641087796562, 130370690653563150, 3385534274596691456, 92801584815121975452, 2677687776095609649256

Offset: 0

Alois P. Heinz, Sep 09 2015

nonn

All terms are even.

Alois P. Heinz, Table of n, a(n) for n = 0..424

Cf. A000670, A032011, A262046, A261982.

g:= proc(n) option remember; `if`(n<2, 1,
       add(binomial(n, k)*g(k), k=0..n-1))
    end:
b:= proc(n, i, p) option remember;
      `if`(i*(i+1)/2n, 0, b(n-i, i-1, p+1)*binomial(n, i))))
    end:
a:= n-> g(n)-b(n$2, 0):
seq(a(n), n=0..25);

g[n_] := g[n] = If[n<2, 1, Sum[Binomial[n, k]*g[k], {k, 0, n-1}]]; b[n_, i_, p_] := b[n, i, p] = If[i*(i+1)/2n, 0, b[n-i, i-1, p+1]*Binomial[n, i]]]]; a[n_] := g[n] - b[n, n, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)

a(n) = A000670(n) - A032011(n).

A374702 Number of integer compositions of n whose leaders of maximal weakly decreasing runs sum to 3. Column k = 3 of A374748.

Original entry on oeis.org

0, 0, 0, 2, 3, 6, 9, 13, 17, 23, 28, 35, 42, 50, 58, 68, 77, 88, 99, 111, 123, 137, 150, 165, 180, 196, 212, 230, 247, 266, 285, 305, 325, 347, 368, 391, 414, 438, 462, 488, 513, 540, 567, 595, 623, 653, 682, 713, 744, 776, 808, 842, 875, 910, 945, 981

Offset: 0

Gus Wiseman, Aug 12 2024

The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.

			The a(0) = 0 through a(8) = 17 compositions:
  .  .  .  (3)   (31)   (32)    (33)     (322)     (332)
           (12)  (112)  (122)   (321)    (331)     (3221)
                 (121)  (311)   (1122)   (1222)    (3311)
                        (1112)  (1221)   (3211)    (11222)
                        (1121)  (3111)   (11122)   (12221)
                        (1211)  (11112)  (11221)   (32111)
                                (11121)  (12211)   (111122)
                                (11211)  (31111)   (111221)
                                (12111)  (111112)  (112211)
                                         (111121)  (122111)
                                         (111211)  (311111)
                                         (112111)  (1111112)
                                         (121111)  (1111121)
                                                   (1111211)
                                                   (1112111)
                                                   (1121111)
                                                   (1211111)

Andrew Howroyd, Table of n, a(n) for n = 0..10000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

The version for k = 2 is A004526.
The version for partitions is A069905 or A001399 (shifted).
For reversed partitions we appear to have A137719.
For length instead of sum we have A241627.
For leaders of constant runs we have A373952.
The opposite rank statistic is A374630, row-sums of A374629.
The corresponding rank statistic is A374741 row-sums of A374740.
Column k = 3 of A374748.
A003242 counts anti-run compositions.
A011782 counts integer compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
Cf. A000009, A000041, A101271, A106356, A188900, A238343, A261982, A333213.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#,GreaterEqual]]==3&]],{n,0,15}]

seq(n)={Vec((2 + x + x^2)/((1 + x + x^2)*(1 + x)*(1 - x)^3) + O(x^(n-2)), -n-1)} \\ Andrew Howroyd, Aug 14 2024

G.f.: x^3*(2 + x + x^2)/((1 + x + x^2)*(1 + x)*(1 - x)^3). - Andrew Howroyd, Aug 14 2024

a(27) onwards from Andrew Howroyd, Aug 14 2024

A335487 Number of (1,1)-matching permutations of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 3, 0, 0, 0, 4, 1, 0, 1, 3, 0, 0, 0, 1, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 3, 3, 0, 0, 5, 1, 3, 0, 3, 0, 4, 0, 4, 0, 0, 0, 12, 0, 0, 3, 1, 0, 0, 0, 3, 0, 0, 0, 10, 0, 0, 3, 3, 0, 0, 0, 5, 1, 0, 0, 12, 0, 0

Offset: 1

Gus Wiseman, Jun 14 2020

nonn

Depends only on sorted prime signature (A118914).
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.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(n) permutations for n = 4, 12, 24, 48, 36, 72, 60:
  (11)  (112)  (1112)  (11112)  (1122)  (11122)  (1123)
        (121)  (1121)  (11121)  (1212)  (11212)  (1132)
        (211)  (1211)  (11211)  (1221)  (11221)  (1213)
               (2111)  (12111)  (2112)  (12112)  (1231)
                       (21111)  (2121)  (12121)  (1312)
                                (2211)  (12211)  (1321)
                                        (21112)  (2113)
                                        (21121)  (2131)
                                        (21211)  (2311)
                                        (22111)  (3112)
                                                 (3121)
                                                 (3211)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Positions of zeros are A005117 (squarefree numbers).
The case where the match must be contiguous is A333175.
The avoiding version is A335489.
The (1,1,1)-matching case is A335510.
Patterns are counted by A000670.
Permutations of prime indices are counted by A008480.
(1,1)-matching patterns are counted by A019472.
(1,1)-matching compositions are counted by A261982.
STC-numbers of permutations of prime indices are A333221.
Patterns matched by standard compositions are counted by A335454.
Dimensions of downsets of standard compositions are A335465.
(1,1)-matching compositions are ranked by A335488.
Cf. A000961, A056239, A056986, A112798, A181796, A335451, A335452, A335460, A335462.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],!UnsameQ@@#&]],{n,100}]

a(n) = 0 if n is squarefree, otherwise a(n) = A008480(n).

a(n) = A008480(n) - A281188(n) for n != 4.

A374254 Numbers k such that the k-th composition in standard order is an anti-run and matches the patterns (1,2,1) or (2,1,2).

Original entry on oeis.org

13, 22, 25, 45, 49, 54, 76, 77, 82, 89, 97, 101, 102, 105, 108, 109, 141, 148, 150, 153, 162, 165, 166, 177, 178, 180, 182, 193, 197, 198, 204, 205, 209, 210, 216, 217, 269, 278, 280, 281, 297, 300, 301, 305, 306, 308, 310, 322, 325, 326, 332, 333, 353, 354

Offset: 1

Gus Wiseman, Jul 14 2024

nonn

Such a composition cannot be strict.

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 their standard compositions begin:
   13: (1,2,1)
   22: (2,1,2)
   25: (1,3,1)
   45: (2,1,2,1)
   49: (1,4,1)
   54: (1,2,1,2)
   76: (3,1,3)
   77: (3,1,2,1)
   82: (2,3,2)
   89: (2,1,3,1)
   97: (1,5,1)
  101: (1,3,2,1)
  102: (1,3,1,2)
  105: (1,2,3,1)
  108: (1,2,1,3)
  109: (1,2,1,2,1)
  141: (4,1,2,1)
  148: (3,2,3)
  150: (3,2,1,2)
  153: (3,1,3,1)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Compositions of this type are counted by A285981.
Permutations of prime indices of this type are counted by A335460.
This is the anti-run complement case of A374249, counted by A274174.
This is the anti-run case of A374253, counted by A335548.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A025047 counts wiggly compositions, ranks A345167.
A066099 lists compositions in standard order.
A124767 counts runs in standard compositions, anti-runs A333381.
A233564 ranks strict compositions, counted by A032020.
A333755 counts compositions by number of runs.
A335454 counts patterns matched by standard compositions.
A335456 counts patterns matched by compositions.
A335462 counts (1,2,1)- and (2,1,2)-matching permutations of prime indices.
A335465 counts minimal patterns avoided by a standard composition.
- A335470 counts (1,2,1)-matching compositions, ranks A335466.
- A335471 counts (1,2,1)-avoiding compositions, ranks A335467.
- A335472 counts (2,1,2)-matching compositions, ranks A335468.
- A335473 counts (2,1,2)-avoiding compositions, ranks A335469.
A373948 encodes run-compression using compositions in standard order.
A373949 counts compositions by run-compressed sum, opposite A373951.
A373953 gives run-compressed sum of standard compositions, excess A373954.
Cf. A106356, A124762, A238130, A238279, A261982, A333175, A333382, A333627, A335463, A335524, A335525.

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

Equals A333489 /\ A374253.

A374703 Number of integer compositions of 2n whose leaders of weakly decreasing runs sum to n. Center n = 2*k of the triangle A374748.

Original entry on oeis.org

1, 1, 2, 9, 24, 96, 343, 1242, 4700, 17352, 65995

Offset: 0

Gus Wiseman, Aug 12 2024

The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.

			The a(0) = 1 through a(4) = 24 compositions:
  ()  (11)  (22)   (33)     (44)
            (211)  (321)    (422)
                   (1122)   (431)
                   (1221)   (1133)
                   (3111)   (1322)
                   (11112)  (1331)
                   (11121)  (4211)
                   (11211)  (11132)
                   (12111)  (11321)
                            (13211)
                            (21122)
                            (21221)
                            (22112)
                            (22121)
                            (41111)
                            (111113)
                            (111131)
                            (111311)
                            (113111)
                            (131111)
                            (211112)
                            (211121)
                            (211211)
                            (212111)

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

For reversed partitions we have A364910.
For strictly decreasing runs we have the center of A374700.
Center n = 2*k of the triangle A374748.
A003242 counts anti-run compositions.
A011782 counts integer compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
Cf. A000041, A004526, A106356, A124766, A188900, A238343, A261982, A333213.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[2n],Total[First/@Split[#,GreaterEqual]]==n&]],{n,0,8}]

cp's OEIS Frontend

A363262 Number of integer compositions of n in which the greatest part appears more than once.

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A374766 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of maximal strictly decreasing runs sum to k.

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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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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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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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A262047 Number of ordered partitions of [n] such that at least two parts have the same size.

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Maple

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A374702 Number of integer compositions of n whose leaders of maximal weakly decreasing runs sum to 3. Column k = 3 of A374748.

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PARI

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A335487 Number of (1,1)-matching permutations of the prime indices of n.

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