A188920 -id:A188920 - cp's OEIS frontend

A374689 Number of integer compositions of n whose leaders of strictly increasing runs are strictly decreasing.

1, 1, 1, 3, 3, 6, 10, 13, 21, 32, 48, 66, 101, 144, 207, 298, 415, 592, 833, 1163, 1615, 2247, 3088, 4259, 5845, 7977, 10862, 14752, 19969, 26941, 36310, 48725, 65279, 87228, 116274, 154660, 205305, 271879, 359400, 474157, 624257, 820450, 1076357, 1409598

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 strictly decreasing. The weakly decreasing version is A374697.

			The a(0) = 1 through a(8) = 21 compositions:
  ()  (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)
                (12)  (13)  (14)   (15)   (16)   (17)
                (21)  (31)  (23)   (24)   (25)   (26)
                            (32)   (42)   (34)   (35)
                            (41)   (51)   (43)   (53)
                            (212)  (123)  (52)   (62)
                                   (213)  (61)   (71)
                                   (231)  (124)  (125)
                                   (312)  (214)  (134)
                                   (321)  (241)  (215)
                                          (313)  (251)
                                          (412)  (314)
                                          (421)  (323)
                                                 (341)
                                                 (413)
                                                 (431)
                                                 (512)
                                                 (521)
                                                 (2123)
                                                 (2312)
                                                 (3212)

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 weak version appears to be A189076.
Ranked by positions of strictly decreasing rows in A374683.
The opposite version is A374762.
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have A000041.
- For leaders of anti-runs we have A374680.
- For leaders of weakly increasing runs we have A188920.
- For leaders of weakly decreasing runs we have A374746.
- For leaders of strictly decreasing runs we have A374763.
Types of run-leaders (instead of strictly decreasing):
- For identical leaders we have A374686, ranks A374685.
- For distinct leaders we have A374687, ranks A374698.
- For strictly increasing leaders we have A374688.
- For weakly increasing leaders we have A374690.
- 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, A238343, A261982, A304969, A333213, A374632, A374635, A374640, A374679.

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

C_x(N) = {my(x='x+O('x^N), h=prod(i=1,N, 1+(x^i)*prod(j=i+1,N, 1+x^j))); Vec(h)}
C_x(50) \\ John Tyler Rascoe, Jul 29 2024

G.f.: Product_{i>0} (1 + (x^i)*Product_{j>i} (1 + x^j)). - John Tyler Rascoe, Jul 29 2024

a(26) onwards from John Tyler Rascoe, Jul 29 2024

A374741 Sum of leaders of weakly decreasing runs in the n-th composition in standard order.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 3, 1, 4, 3, 2, 2, 4, 3, 3, 1, 5, 4, 3, 3, 5, 2, 4, 2, 5, 4, 3, 3, 4, 3, 3, 1, 6, 5, 4, 4, 3, 3, 5, 3, 6, 5, 2, 2, 5, 4, 4, 2, 6, 5, 4, 4, 6, 3, 5, 3, 5, 4, 3, 3, 4, 3, 3, 1, 7, 6, 5, 5, 4, 4, 6, 4, 7, 3, 3, 3, 6, 5, 5, 3, 7, 6, 5, 5, 5, 2, 4

Offset: 0

Gus Wiseman, Jul 24 2024

nonn

The leaders of weakly decreasing runs in a sequence are obtained by splitting it into maximal weakly decreasing 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 maximal weakly decreasing subsequences of the 1234567th composition in standard order are ((3,2,1),(2,2,1),(2),(5,1,1,1)), so a(1234567) is 3+2+2+5 = 12.

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

For length instead of sum we have A124765.
Other types of runs are A373953, A374516, A374684, A374758.
The opposite is A374630.
Row-sums of A374740, opposite A374629.
Counting compositions by this statistic gives A374748, opposite A374637.
A373949 counts compositions by run-compressed sum.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) (or sometimes A070939).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of strict compositions are A233564, counted by A032020.
- Constant compositions are ranked by A272919.
- Ranks of anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627.
- Run-compression transform is A373948.
Cf. A065120, A106356, A188920, A189076, A238343, A333175, A333213, A373949, A373954, A374631, A374633, A374635, A374701, A374742.

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

A374758 Sum of leaders of strictly decreasing runs in the n-th composition in standard order.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 3, 4, 3, 4, 4, 5, 4, 3, 4, 5, 4, 4, 4, 5, 4, 5, 4, 5, 4, 5, 5, 6, 5, 4, 5, 6, 3, 5, 5, 6, 5, 6, 5, 5, 4, 5, 5, 6, 5, 4, 5, 6, 5, 5, 5, 6, 5, 6, 5, 6, 5, 6, 6, 7, 6, 5, 6, 4, 4, 6, 6, 7, 6, 5, 4, 6, 5, 6, 6, 7, 6, 5, 6, 7, 6, 6

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.

			The maximal strictly decreasing subsequences of the 1234567th composition in standard order are ((3,2,1),(2),(2,1),(2),(5,1),(1),(1)) with leaders (3,2,2,2,5,1,1), so a(1234567) = 16.

Gus Wiseman, Four statistics for runs and antiruns of prime, nonprime, squarefree, and nonsquarefree numbers.

Row sums of A374757.
For leaders of constant runs we have A373953.
For leaders of anti-runs we have A374516.
For leaders of weakly increasing runs we have A374630.
For length instead of sum we have A124769.
The opposite version is A374684, sum of A374683 (length A124768).
The case of partitions ranked by Heinz numbers is A374706.
The weak version is A374741, sum of A374740 (length A124765).
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.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Ranks of non-contiguous compositions are A374253, counted by A335548.
Six types of runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A106356, A188920, A189076, A233564, A238343, A272919, A333213, A373949, A374700, A374759, A374761, A374767.

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

A374688 Number of integer compositions of n whose leaders of strictly increasing runs are themselves strictly increasing.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 7, 11, 16, 21, 31, 45, 63, 87, 122, 170, 238, 328, 449, 616, 844, 1151, 1565, 2121, 2861, 3855, 5183, 6953, 9299, 12407, 16513, 21935, 29078, 38468, 50793, 66935, 88037, 115577, 151473, 198175, 258852, 337560, 439507, 571355, 741631

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 strictly decreasing.

			The a(0) = 1 through a(9) = 16 compositions:
  ()  (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)     (9)
                (12)  (13)  (14)   (15)   (16)   (17)    (18)
                            (23)   (24)   (25)   (26)    (27)
                            (122)  (123)  (34)   (35)    (36)
                                   (132)  (124)  (125)   (45)
                                          (133)  (134)   (126)
                                          (142)  (143)   (135)
                                                 (152)   (144)
                                                 (233)   (153)
                                                 (1223)  (162)
                                                 (1232)  (234)
                                                         (243)
                                                         (1224)
                                                         (1233)
                                                         (1242)
                                                         (1323)

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).

The weak version is A374635.
Ranked by positions of strictly increasing rows in A374683 (sums A374684).
The opposite version is A374763.
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have A000041.
- For leaders of anti-runs we have A374679.
- For leaders of weakly increasing runs we have A374634.
- For leaders of strictly decreasing runs we have A374762.
Types of run-leaders (instead of strictly increasing):
- For identical leaders we have A374686, ranks A374685.
- For distinct leaders we have A374687, ranks A374698.
- For strictly decreasing leaders we have A374689.
- For weakly increasing leaders we have A374690.
- 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.
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, A374632.

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

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

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

A374684 Sum of leaders of strictly increasing runs in the n-th composition in standard order.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 1, 3, 4, 4, 4, 4, 1, 2, 2, 4, 5, 5, 5, 5, 2, 5, 3, 5, 1, 2, 3, 3, 2, 3, 3, 5, 6, 6, 6, 6, 6, 6, 4, 6, 2, 3, 6, 6, 3, 4, 4, 6, 1, 2, 3, 3, 1, 4, 2, 4, 2, 3, 4, 4, 3, 4, 4, 6, 7, 7, 7, 7, 7, 7, 5, 7, 3, 7, 7, 7, 4, 5, 5, 7, 2, 3, 4, 4, 4, 7, 5

Offset: 0

Gus Wiseman, Jul 26 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 maximal strictly increasing subsequences of the 1234567th composition in standard order are ((3),(2),(1,2),(2),(1,2,5),(1),(1),(1)) with leaders (3,2,1,2,1,1,1,1), so a(1234567) = 12.

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

The weak version is A374630.
Row-sums of A374683.
The opposite version is A374758.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) (or sometimes A070939).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Run-length transform is A333627.
- Run-compression transform is A373948.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Ranks of non-contiguous compositions are A374253, counted by A335548.
Cf. A065120, A106356, A188920, A189076, A238343, A272919, A374634, A374685, A374698.
Cf. A374251 (sums A373953), A374515 (sums A374516), A374740 (sums A374741).

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

A374678 Number of integer compositions of n whose leaders of maximal anti-runs are not distinct.

Original entry on oeis.org

0, 0, 1, 1, 3, 7, 15, 32, 70, 144, 311, 653, 1354, 2820, 5850, 12054, 24810, 50923, 104206, 212841, 433919, 882930, 1793810, 3639248, 7373539, 14921986

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 anti-runs of y = (1,1,2,2) are ((1),(1,2),(2)) with leaders (1,1,2) so y is counted under a(6).
The a(0) = 0 through a(6) = 15 compositions:
  .  .  (11)  (111)  (22)    (113)    (33)
                     (112)   (221)    (114)
                     (1111)  (1112)   (222)
                             (1121)   (1113)
                             (1211)   (1122)
                             (2111)   (1131)
                             (11111)  (1311)
                                      (2211)
                                      (3111)
                                      (11112)
                                      (11121)
                                      (11211)
                                      (12111)
                                      (21111)
                                      (111111)

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

For constant runs we have A335548, complement A274174, ranks A374249.
The complement is counted by A374518, ranks A374638.
For weakly increasing runs we have complement A374632, ranks A374768.
Compositions of this type are ranked by A374639.
For identical instead of distinct leaders we have A374640, ranks A374520, complement A374517, ranks A374519.
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, A239955, A333213, A373949, A374515.

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

A374680 Number of integer compositions of n whose leaders of anti-runs are strictly decreasing.

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 16, 31, 52, 98, 179, 323, 590, 1078, 1945, 3531, 6421, 11621, 21041, 38116, 68904, 124562, 225138, 406513, 733710, 1323803

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(6) = 16 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)
                (12)  (13)   (14)   (15)
                (21)  (31)   (23)   (24)
                      (121)  (32)   (42)
                      (211)  (41)   (51)
                             (131)  (123)
                             (212)  (132)
                             (311)  (141)
                                    (213)
                                    (231)
                                    (312)
                                    (321)
                                    (411)
                                    (1212)
                                    (2112)
                                    (2121)

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

For distinct but not necessarily decreasing leaders we have A374518.
For partitions instead of compositions we have A375133.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we have A188920.
- For leaders of weakly decreasing runs we have A374746.
- For leaders of strictly decreasing runs we have A374763.
- For leaders of strictly increasing runs we have A374689.
Other types of run-leaders (instead of strictly 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 weakly decreasing leaders we have A374682.
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. A189076, A238343, A333213, A333381, A373949, A374515, A374632, A374678, A374700, A374706.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Greater@@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

A374747 Number of integer compositions of n whose leaders of weakly decreasing runs are themselves weakly decreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 14, 24, 43, 76, 136, 242, 431, 764, 1353, 2387, 4202, 7376, 12918, 22567, 39338, 68421, 118765, 205743, 355756, 614038, 1058023, 1820029, 3125916, 5360659, 9179700, 15697559, 26807303, 45720739, 77881393, 132505599, 225182047, 382252310, 648187055

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 composition y = (3,2,1,2,2,1,2,5,1,1,1) has weakly decreasing runs ((3,2,1),(2,2,1),(2),(5,1,1,1)), with leaders (3,2,2,5), which are not weakly decreasing, so y is not counted under a(21).
The a(0) = 1 through a(6) = 14 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)
                                         (2112)
                                         (2121)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

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

Ranked by positions of weakly decreasing rows in A374740, opposite A374629.
Types of runs (instead of weakly 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 strictly decreasing runs we have A374765.
Types of run-leaders (instead of weakly decreasing):
- 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 increasing leaders we have opposite A374634.
- For strictly decreasing leaders we have A374746.
A011782 counts compositions.
A124765 counts weakly decreasing runs in standard 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, A238343, A261982, A333213, A374630, A374635, A374636, A374741.

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

dfs(m, r, u) = 1 + sum(s=r+1, min(m, u), x^s/(1-x^s) + sum(t=1, min(s-1, m-s), dfs(m-s-t, t, s)*x^(s+t)/prod(i=t, s, 1-x^i)));
lista(nn) = Vec(dfs(nn, 0, nn) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 14 2025

More terms from Jinyuan Wang, Feb 14 2025

cp's OEIS Frontend

A374689 Number of integer compositions of n whose leaders of strictly increasing runs are strictly decreasing.

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A374741 Sum of leaders of weakly decreasing runs in the n-th composition in standard order.

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A374758 Sum of leaders of strictly decreasing runs in the n-th composition in standard order.

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A374688 Number of integer compositions of n whose leaders of strictly increasing runs are themselves strictly increasing.

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

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A374684 Sum of leaders of strictly increasing runs in the n-th composition in standard order.

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A374678 Number of integer compositions of n whose leaders of maximal anti-runs are not distinct.

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A374680 Number of integer compositions of n whose leaders of anti-runs are strictly decreasing.

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