A374697 -id:A374697 - cp's OEIS frontend

A374687 Number of integer compositions of n whose leaders of strictly increasing runs are distinct.

1, 1, 1, 3, 3, 7, 11, 15, 27, 45, 65, 101, 161, 251, 381, 573, 865, 1321, 1975, 2965, 4387, 6467, 9579, 14091, 20669, 30135, 43869, 63531, 91831, 132575, 190567, 273209, 390659, 557069, 792371, 1124381, 1591977, 2249029, 3169993, 4458163, 6256201, 8762251, 12246541

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 a(0) = 1 through a(7) = 15 compositions:
  ()  (1)  (2)  (3)   (4)   (5)    (6)    (7)
                (12)  (13)  (14)   (15)   (16)
                (21)  (31)  (23)   (24)   (25)
                            (32)   (42)   (34)
                            (41)   (51)   (43)
                            (122)  (123)  (52)
                            (212)  (132)  (61)
                                   (213)  (124)
                                   (231)  (133)
                                   (312)  (142)
                                   (321)  (214)
                                          (241)
                                          (313)
                                          (412)
                                          (421)

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

Ranked by A374698.
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have A274174 for n > 0, 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 weakly decreasing runs we have A374743, ranks A374701.
- For leaders of strictly decreasing runs we have A374761, ranks A374767.
Types of run-leaders (instead of distinct):
- For identical leaders we have A374686, ranks A374685.
- For strictly increasing leaders we have A374688.
- 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.
A335456 counts patterns matched by compositions.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374683 lists leaders of strictly increasing runs of standard compositions.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A000009, A106356, A188920, A189076, A238343, A333213, A335548, A374629, A374634, A374635.

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

dfs(m, r, v) = 1 + sum(s=1, min(m, r), if(!setsearch(v, s), dfs(m-s, s, setunion(v, [s]))*x^s + sum(t=s+1, m-s, dfs(m-s-t, t, setunion(v, [s]))*x^(s+t)*prod(i=s+1, t-1, 1+x^i))));
lista(nn) = Vec(dfs(nn, nn, []) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

A374686 Number of integer compositions of n whose leaders of strictly increasing runs are identical.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 17, 29, 51, 91, 162, 291, 523, 948, 1712, 3112, 5656, 10297, 18763, 34217, 62442, 114006, 208239, 380465, 695342, 1271046, 2323818, 4249113, 7770389, 14210991, 25991853, 47541734, 86962675, 159077005, 291001483, 532345978, 973871397

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 identical. For maxima instead of minima we have A374760. For all partitions (not just strict) we have A374704, for maxima A358905.

			The composition (2,3,2,2,3,4) has strictly increasing runs ((2,3),(2),(2,3,4)), with leaders (2,2,2), so is counted under a(16).
The a(0) = 1 through a(6) = 17 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (12)   (13)    (14)     (15)
                 (111)  (22)    (23)     (24)
                        (112)   (113)    (33)
                        (121)   (131)    (114)
                        (1111)  (1112)   (123)
                                (1121)   (141)
                                (1211)   (222)
                                (11111)  (1113)
                                         (1131)
                                         (1212)
                                         (1311)
                                         (11112)
                                         (11121)
                                         (11211)
                                         (12111)
                                         (111111)

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 A374685.
Types of runs (instead of strictly increasing):
- 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 weakly increasing runs we have A374631, ranks A374633.
- For leaders of weakly decreasing runs we have A374742, ranks A374744.
- For leaders of strictly decreasing runs we have A374760, ranks A374759.
Types of run-leaders (instead of identical):
- For distinct leaders we have A374687, ranks A374698.
- For strictly increasing leaders we have A374688.
- 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.
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.
A374683 lists leaders of strictly increasing runs of standard compositions.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A000009, A106356, A188920, A189076, A238343, A304969, A333213, A374632, A374634, A374635, A374640.

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

seq(n) = Vec(1 + sum(k=1, n, 1/(1 - x^k*prod(j=k+1, n-k, 1 + x^j, 1 + O(x^(n-k+1))))-1)) \\ Andrew Howroyd, Jul 27 2024

a(26) onwards from Andrew Howroyd, Jul 27 2024

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

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

Original entry on oeis.org

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

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

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}]

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

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

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

cp's OEIS Frontend

A374687 Number of integer compositions of n whose leaders of strictly increasing runs are distinct.

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A374686 Number of integer compositions of n whose leaders of strictly increasing runs are identical.

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

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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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A374682 Number of integer compositions of n whose leaders of anti-runs are weakly decreasing.

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

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