A374679 -id:A374679 - cp's OEIS frontend

A374681 Number of integer compositions of n whose leaders of anti-runs are weakly increasing.

1, 1, 2, 4, 7, 14, 27, 50, 96, 185, 353, 672, 1289, 2466, 4722, 9052, 17342, 33244, 63767, 122325, 234727, 450553, 864975, 1660951, 3190089, 6128033

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) = 14 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)
                                (1211)
                                (11111)

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

For partitions instead of compositions we have A034296.
Other types of runs (instead of anti-):
- For leaders of constant runs we have A000041.
- For leaders of weakly decreasing runs we have A188900.
- For leaders of weakly increasing runs we have A374635.
- For leaders of strictly increasing runs we have A374690.
- For leaders of strictly decreasing runs we have A374764.
Other types of run-leaders (instead of weakly increasing):
- For identical leaders we have A374517, ranks A374519.
- For distinct leaders we have A374518, ranks A374638.
- For strictly increasing leaders we have A374679.
- For weakly decreasing leaders we have A374682.
- 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. A189076, A238343, A333213, A333381, A373949, A374515, A374632, A374678, A374700, A374706.

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

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}]

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 29 2024

nonn

The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.

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

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

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

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

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

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

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

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

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

A375135 Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 9, 25, 63, 152, 355, 809, 1804, 3963, 8590, 18423, 39161, 82620, 173198, 361101, 749326, 1548609, 3189132, 6547190, 13404613, 27378579, 55801506, 113517749, 230544752, 467519136, 946815630, 1915199736, 3869892105, 7812086380, 15756526347

Offset: 0

Gus Wiseman, Aug 06 2024

nonn

The leaders of maximal strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

			The composition y = (1,2,1,3,2,3) has strictly increasing runs ((1,2),(1,3),(2,3)), with leaders (1,1,2), which are not weakly decreasing, so y is counted under a(12).
The a(0) = 0 through a(8) = 25 compositions:
  .  .  .  .  .  (122)  (132)   (133)    (143)
                        (1122)  (142)    (152)
                        (1221)  (1132)   (233)
                                (1222)   (1133)
                                (1321)   (1142)
                                (2122)   (1223)
                                (11122)  (1232)
                                (11221)  (1322)
                                (12211)  (1331)
                                         (1421)
                                         (2132)
                                         (3122)
                                         (11132)
                                         (11222)
                                         (11321)
                                         (12122)
                                         (12212)
                                         (12221)
                                         (13211)
                                         (21122)
                                         (21221)
                                         (111122)
                                         (111221)
                                         (112211)
                                         (122111)

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

For leaders of constant runs we have A056823.
For leaders of weakly increasing runs we have A374636, complement A189076?
The complement is counted by A374697.
For leaders of anti-runs we have A374699, complement A374682.
Other functional neighbors: A188920, A374764, A374765.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A106356, A238343, A261982, A333213, A374632, A374679, A374683, A374689.

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

a(n) = A011782(n) - A374697(n). - Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

A374699 Number of integer compositions of n whose leaders of maximal anti-runs are not weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 5, 14, 34, 78, 180, 407, 907, 2000, 4364, 9448, 20323, 43448, 92400, 195604, 412355, 866085, 1813035, 3783895, 7875552

Offset: 0

Gus Wiseman, Aug 06 2024

The leaders of maximal anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

			The a(0) = 0 through a(8) = 14 compositions:
  .  .  .  .  .  (122)  (1122)  (133)    (233)
                        (1221)  (1222)   (1133)
                                (11122)  (1223)
                                (11221)  (1322)
                                (12211)  (1331)
                                         (11222)
                                         (12122)
                                         (12212)
                                         (12221)
                                         (21122)
                                         (111122)
                                         (111221)
                                         (112211)
                                         (122111)

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

The complement is counted by A374682.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A056823.
- For leaders of weakly increasing runs we have A374636, complement A189076?
- For leaders of strictly increasing runs: A375135, complement A374697.
Other types of run-leaders (instead of weakly decreasing):
- For identical leaders we have A374640, ranks A374520, complement A374517, ranks A374519.
- For distinct leaders we have A374678, ranks A374639, complement A374518, ranks A374638.
- For weakly increasing leaders we have complement A374681.
- For strictly increasing leaders we have complement complement A374679.
- For strictly decreasing leaders we have complement 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.
A333381 counts maximal anti-runs in standard compositions.
Cf. A115029, A238343, A333213, A373949, A374515, A374632, A374635, A374700.

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

A375406 Number of integer compositions of n that match the dashed pattern 3-12.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 4, 14, 41, 110, 278, 673, 1576, 3599, 8055, 17732, 38509, 82683, 175830, 370856, 776723, 1616945, 3348500, 6902905, 14174198, 29004911, 59175625, 120414435, 244468774, 495340191, 1001911626, 2023473267, 4081241473, 8222198324, 16548146045, 33276169507

Offset: 0

Gus Wiseman, Aug 22 2024

nonn

First differs from the non-dashed version A335514 at a(9) = 41, A335514(9) = 42, due to the composition (3,1,3,2).

Also the number of integer compositions of n whose leaders of weakly decreasing runs are not weakly increasing. For example, the composition q = (1,1,2,1,2,2,1,3) has maximal weakly decreasing runs ((1,1),(2,1),(2,2,1),(3)), with leaders (1,2,2,3), which are weakly increasing, so q is not counted under a(13); also q does not match 3-12. On the other hand, the reverse is (3,1,2,2,1,2,1,1), with maximal weakly decreasing runs ((3,1),(2,2,1),(2,1,1)), with leaders (3,2,2), which are not weakly increasing, so it is counted under a(13); meanwhile it matches 3-12, as required.

			The a(0) = 0 through a(8) = 14 compositions:
  .  .  .  .  .  .  (312)  (412)   (413)
                           (1312)  (512)
                           (3112)  (1412)
                           (3121)  (2312)
                                   (3122)
                                   (3212)
                                   (4112)
                                   (4121)
                                   (11312)
                                   (13112)
                                   (13121)
                                   (31112)
                                   (31121)
                                   (31211)

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

For leaders of identical runs we have A056823.
The complement is counted by A188900.
The non-dashed version is A335514, ranks A335479.
Ranks are positions of non-weakly increasing rows in A374740.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
Counting compositions by number of runs: A238130, A238279, A333755.
A373949 counts compositions by run-compressed sum, opposite A373951.
Cf. A106356, A188920, A189076, A189077, A238343, A333213, A335548, A374629, A374637, A374679, A374748.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !LessEqual@@First/@Split[#,GreaterEqual]&]],{n,0,15}]
- or -
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MatchQ[#,{_,z_,_,x_,y_,_}/;x

a(n>0) = 2^(n-1) - A188900(n).

cp's OEIS Frontend

A374681 Number of integer compositions of n whose leaders of anti-runs are weakly increasing.

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

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

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

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

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A375135 Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.

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A374699 Number of integer compositions of n whose leaders of maximal anti-runs are not weakly decreasing.

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