A374747 -id:A374747 - cp's OEIS frontend

A358836 Number of multiset partitions of integer partitions of n with all distinct block sizes.

1, 1, 2, 4, 8, 15, 28, 51, 92, 164, 289, 504, 871, 1493, 2539, 4290, 7201, 12017, 19939, 32911, 54044, 88330, 143709, 232817, 375640, 603755, 966816, 1542776, 2453536, 3889338, 6146126, 9683279, 15211881, 23830271, 37230720, 58015116, 90174847, 139820368, 216286593

Offset: 0

Gus Wiseman, Dec 05 2022

nonn

Also the number of integer compositions of n whose leaders of maximal weakly decreasing runs are strictly increasing. For example, the composition (1,2,2,1,3,1,4,1) has maximal weakly decreasing runs ((1),(2,2,1),(3,1),(4,1)), with leaders (1,2,3,4), so is counted under a(15). - Gus Wiseman, Aug 21 2024

			The a(1) = 1 through a(5) = 15 multiset partitions:
  {1}  {2}    {3}        {4}          {5}
       {1,1}  {1,2}      {1,3}        {1,4}
              {1,1,1}    {2,2}        {2,3}
              {1},{1,1}  {1,1,2}      {1,1,3}
                         {1,1,1,1}    {1,2,2}
                         {1},{1,2}    {1,1,1,2}
                         {2},{1,1}    {1},{1,3}
                         {1},{1,1,1}  {1},{2,2}
                                      {2},{1,2}
                                      {3},{1,1}
                                      {1,1,1,1,1}
                                      {1},{1,1,2}
                                      {2},{1,1,1}
                                      {1},{1,1,1,1}
                                      {1,1},{1,1,1}
From _Gus Wiseman_, Aug 21 2024: (Start)
The a(0) = 1 through a(5) = 15 compositions whose leaders of maximal weakly decreasing runs are strictly increasing:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (122)
                        (1111)  (131)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)
(End)

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 version for set partitions is A007837.
For sums instead of sizes we have A271619.
For constant instead of distinct sizes we have A319066.
These multiset partitions are ranked by A326533.
For odd instead of distinct sizes we have A356932.
The version for twice-partitions is A358830.
The case of distinct sums also is A358832.
Ranked by positions of strictly increasing rows in A374740, opposite A374629.
A001970 counts multiset partitions of integer partitions.
A011782 counts compositions.
A063834 counts twice-partitions, strict A296122.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
Runs and leaders: A000041, A188900, A374634, A374635, A374637, A374679, A374742 (ranks A374744), A374743 (ranks A374701), A374746, A374747.
Cf. A000009, A141199, A273873, A279375, A279785, A279790, A358334.
Cf. A003242, A106356, A188920, A189076, A238343, A333213, A336342.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],UnsameQ@@Length/@#&]],{n,0,10}]
(* second program *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Less@@First/@Split[#,GreaterEqual]&]],{n,0,15}] (* Gus Wiseman, Aug 21 2024 *)

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=P(n,y)); Vec(prod(k=1, n, 1 + polcoef(g, k, y) + O(x*x^n)))} \\ Andrew Howroyd, Dec 31 2022

G.f.: Product_{k>=1} (1 + [y^k]P(x,y)) where P(x,y) = 1/Product_{k>=1} (1 - y*x^k). - Andrew Howroyd, Dec 31 2022

Terms a(11) and beyond from Andrew Howroyd, Dec 31 2022

A374742 Number of integer compositions of n whose leaders of weakly decreasing runs are identical.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 54, 87, 138, 220, 349, 556, 881, 1403, 2229, 3551, 5653, 9019, 14387, 22988, 36739, 58785, 94100, 150765, 241658, 387617, 622002, 998658, 1604032, 2577512, 4143243, 6662520, 10716931, 17243904, 27753518, 44680121, 71947123, 115880662

Offset: 0

Gus Wiseman, Jul 25 2024

nonn

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

			The composition (3,1,3,2,1,3,3) has maximal weakly decreasing subsequences ((3,1),(3,2,1),(3,3)), with leaders (3,3,3), so is counted under a(16).
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)    (321)
                                (2111)   (411)
                                (11111)  (2112)
                                         (2121)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

John Tyler Rascoe, Table of n, a(n) for n = 0..200
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Ranked by A374744 = positions of identical rows in A374740, cf. A374629.
Types of runs (instead of weakly decreasing):
- For leaders of identical runs we have A000005 for n > 0, ranks A272919.
- For leaders of anti-runs we have A374517, ranks A374519.
- For leaders of strictly increasing runs we have A374686, ranks A374685.
- For leaders of weakly increasing runs we have A374631, ranks A374633.
- For leaders of strictly decreasing runs we have A374760, ranks A374759.
Types of run-leaders (instead of identical):
- For strictly decreasing leaders we have A374746.
- For weakly decreasing leaders we have A374747.
- For distinct leaders we have A374743, ranks A374701.
- For weakly increasing leaders we appear to have A188900.
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.
A335456 counts patterns matched by compositions.
A335548 counts non-contiguous compositions, ranks A374253.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
Cf. A000009, A106356, A188920, A189076, A238343, A261982, A333213, A374632, A374634, A374635, A374741.

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

B(i) = x^i/(1-x^i) * sum(j=1,i-1, x^j*prod(k=1,j, (1-x^k)^(-1)))
A_x(N) = {my(x='x+O('x^N)); Vec(1+sum(i=1,N,-1+(1+x^i/(1-x^i))/(1-B(i))))}
A_x(30) \\ John Tyler Rascoe, Apr 29 2025

G.f.: 1 + Sum_{i>0} -1 + (1 + x^i/(1 - x^i))/(1 - B(i,x)) where B(i,x) = x^i/(1 - x^i) * Sum_{j=1..i-1} x^j * Product_{k=1..j} (1 - x^k)^(-1). - John Tyler Rascoe, Apr 29 2025

a(24)-a(40) from Alois P. Heinz, Jul 26 2024

A374743 Number of integer compositions of n whose leaders of weakly decreasing runs are distinct.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 29, 55, 105, 198, 371, 690, 1280, 2364, 4353, 7981, 14568, 26466, 47876, 86264, 154896, 277236, 494675, 879924, 1560275, 2757830, 4859010, 8534420, 14945107, 26096824, 45446624, 78939432, 136773519, 236401194, 407614349, 701147189, 1203194421

Offset: 0

Gus Wiseman, Jul 25 2024

nonn

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 composition (1,3,1,4,1,2,2,1) has maximal weakly decreasing subsequences ((1),(3,1),(4,1),(2,2,1)), with leaders (1,3,4,2), so is counted under a(15).
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)   (122)
                        (1111)  (131)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)

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

Ranked by A374701 = positions of distinct rows in A374740, opposite A374629.
Types of runs (instead of weakly decreasing):
- For leaders of identical runs we have A274174, ranks A374249.
- For leaders of anti-runs we have A374518, ranks A374638.
- For leaders of weakly increasing runs we have A374632, ranks A374768.
- For leaders of strictly increasing runs we have A374687, ranks A374698.
- For leaders of strictly decreasing runs we have A374761, ranks A374767.
Types of run-leaders (instead of distinct):
- For weakly increasing leaders we appear to have A188900.
- For identical leaders we have A374742.
- For strictly increasing leaders we have opposite A374634.
- For strictly decreasing leaders we have A374746.
- For weakly decreasing leaders we have A374747.
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.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
Cf. A000009, A003242, A106356, A188920, A189076, A238343, A261982, A333213, A374635, A374741.

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

a(24)-a(36) from Alois P. Heinz, Jul 26 2024

A374748 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of weakly decreasing runs sum to k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 3, 2, 0, 1, 2, 6, 4, 3, 0, 1, 3, 9, 8, 7, 4, 0, 1, 3, 13, 15, 16, 11, 5, 0, 1, 4, 17, 24, 32, 28, 16, 6, 0, 1, 4, 23, 36, 58, 58, 44, 24, 8, 0, 1, 5, 28, 52, 96, 115, 100, 71, 34, 10, 0, 1, 5, 35, 72, 151, 203, 211, 176, 109, 49, 12

Offset: 0

Gus Wiseman, Jul 26 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.

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   1   2
   0   1   2   3   2
   0   1   2   6   4   3
   0   1   3   9   8   7   4
   0   1   3  13  15  16  11   5
   0   1   4  17  24  32  28  16   6
   0   1   4  23  36  58  58  44  24   8
   0   1   5  28  52  96 115 100  71  34  10
   0   1   5  35  72 151 203 211 176 109  49  12
Row n = 6 counts the following compositions:
  .  (111111)  (222)    (33)     (42)    (51)    (6)
               (2211)   (321)    (411)   (141)   (15)
               (21111)  (3111)   (132)   (114)   (24)
                        (1221)   (1311)  (312)   (123)
                        (1122)   (1131)  (231)
                        (12111)  (1113)  (213)
                        (11211)  (2121)  (1212)
                        (11121)  (2112)
                        (11112)

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

Column n = k is A000009.
Column k = 2 is A004526.
Row-sums are A011782.
For length instead of sum we have A238343.
The opposite rank statistic is A374630, row-sums of A374629.
Column k = 3 is A374702.
The center n = 2k is A374703.
The corresponding rank statistic is A374741 row-sums of A374740.
Types of runs (instead of weakly decreasing):
- For leaders of constant 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 strictly decreasing runs we have A374766.
Types of run-leaders:
- For weakly increasing leaders we appear to have A188900.
- For identical leaders we have A374742, ranks A374744.
- For distinct leaders we have A374743, ranks A374701.
- For strictly decreasing leaders we have A374746.
- For weakly decreasing leaders we have A374747.
A003242 counts anti-run compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A335456 counts patterns matched by compositions.
A335548 counts non-contiguous compositions, ranks A374253.
Cf. A000041, A106356, A124766, A261982, A333213, A374251, A374761.

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

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

A374697 Number of integer compositions of n whose leaders of strictly increasing runs are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 29, 55, 103, 193, 360, 669, 1239, 2292, 4229, 7794, 14345, 26375, 48452, 88946, 163187, 299250, 548543, 1005172, 1841418, 3372603, 6175853, 11307358, 20699979, 37890704, 69351776, 126926194, 232283912, 425075191, 777848212, 1423342837, 2604427561

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.

Also the number of ways to choose a strict integer partition of each part of an integer composition of n (A304969) such that the minima are weakly decreasing [weakly increasing works too].

			The composition (1,2,1,3,2,3) has strictly increasing runs ((1,2),(1,3),(2,3)), with leaders (1,1,2), so is not counted under a(12).
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)

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 A374764.
Ranked by positions of weakly decreasing rows in A374683.
Interchanging weak/strict appears to give A188920, opposite A358836.
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have A000041.
- For leaders of anti-runs we have A374682.
- 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.
Types of run-leaders (instead of weakly decreasing):
- For identical leaders we have A374686, ranks A374685.
- For distinct leaders we have A374687, ranks A374698.
- For weakly increasing leaders we have A374690.
- For strictly increasing leaders we have A374688.
- For strictly decreasing leaders we have A374689.
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, A238343, A261982, A333213, A374632, A374679, A374740.

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

seq(n) = Vec(1/prod(k=1, n, 1 - x^k*prod(j=k+1, n-k, 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>=k+1} (1 + x^j))). - Andrew Howroyd, Jul 31 2024

a(26) onwards from Andrew Howroyd, Jul 31 2024

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}]

A374746 Number of integer compositions of n whose leaders of weakly decreasing runs are strictly decreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 18, 31, 51, 86, 143, 241, 397, 657, 1082, 1771, 2889, 4697, 7605, 12269, 19720, 31580, 50412, 80205, 127208, 201149, 317171, 498717, 782076, 1223230, 1908381, 2969950, 4610949, 7141972, 11037276, 17019617, 26188490, 40213388, 61624824

Offset: 0

Gus Wiseman, Jul 26 2024

nonn

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(7) = 18 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)
           (11)  (21)   (22)    (32)     (33)      (43)
                 (111)  (31)    (41)     (42)      (52)
                        (211)   (221)    (51)      (61)
                        (1111)  (311)    (222)     (322)
                                (2111)   (312)     (331)
                                (11111)  (321)     (412)
                                         (411)     (421)
                                         (2211)    (511)
                                         (3111)    (2221)
                                         (21111)   (3112)
                                         (111111)  (3121)
                                                   (3211)
                                                   (4111)
                                                   (22111)
                                                   (31111)
                                                   (211111)
                                                   (1111111)

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).

Ranked by positions of strictly decreasing rows in A374740, opp. A374629.
Types of runs (instead of weakly decreasing):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we have A188920.
- For leaders of anti-runs we have A374680.
- For leaders of strictly increasing runs we have A374689.
- For leaders of strictly decreasing runs we have A374763.
Types of run-leaders (instead of strictly decreasing):
- For weakly increasing leaders we appear to have A188900.
- For identical leaders we have A374742.
- For distinct leaders we have A374743, ranks A374701.
- For strictly increasing leaders we have opposite A374634.
- For weakly decreasing leaders we have A374747.
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.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
Cf. A000009, A003242, A106356, A189076, A238343, A261982, A333213, A358836, A374632, A374635, A374741.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Greater@@First/@Split[#,GreaterEqual]&]],{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/(1-x^k); p /= 1 - x^k; q *= (1 - x^k/(1-x^k) + x^k*p)/(1-x^k) );  Vec(r + x^(n\2+1)*q/(1-x))} \\ Andrew Howroyd, Dec 30 2024

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

a(24)-a(39) from Alois P. Heinz, Jul 26 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

cp's OEIS Frontend

A358836 Number of multiset partitions of integer partitions of n with all distinct block sizes.

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A374742 Number of integer compositions of n whose leaders of weakly decreasing runs are identical.

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A374743 Number of integer compositions of n whose leaders of weakly decreasing runs are distinct.

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A374748 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of weakly decreasing runs sum to k.

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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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A374697 Number of integer compositions of n whose leaders of strictly increasing runs are 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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