A374687 -id:A374687 - cp's OEIS frontend

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

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

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

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

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

A374521 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of anti-runs sum to k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 1, 1, 2, 0, 2, 1, 2, 3, 0, 2, 5, 3, 4, 2, 0, 5, 7, 8, 3, 5, 4, 0, 9, 12, 11, 17, 5, 8, 2, 0, 14, 26, 23, 22, 24, 6, 9, 4, 0, 25, 42, 54, 41, 36, 36, 7, 12, 3, 0, 46, 76, 88, 107, 60, 60, 48, 9, 14, 4

Offset: 0

Gus Wiseman, Aug 02 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.

			Triangle begins:
   1
   0   1
   0   0   2
   0   1   1   2
   0   2   1   2   3
   0   2   5   3   4   2
   0   5   7   8   3   5   4
   0   9  12  11  17   5   8   2
   0  14  26  23  22  24   6   9   4
   0  25  42  54  41  36  36   7  12   3
   0  46  76  88 107  60  60  48   9  14   4
   0  78 144 166 179 176 101  83  68  10  17   2
   0 136 258 327 339 311 299 139 122  81  12  18   6
   0 242 457 602 704 591 544 447 198 165 109  12  23   2
Row n = 6 counts the following compositions:
  .  (15)    (24)    (321)    (42)     (51)     (6)
     (141)   (114)   (312)    (1122)   (411)    (33)
     (132)   (231)   (1113)   (11112)  (3111)   (222)
     (123)   (213)   (2112)            (2211)   (111111)
     (1212)  (1311)  (1221)            (21111)
             (1131)  (12111)
             (2121)  (11211)
                     (11121)

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

Column n = k is A000005, except a(0) = 1.
Row-sums are A011782.
Column k = 1 is A096569.
For length instead of sum we have A106356.
The corresponding rank statistic is A374516, row-sums of A374515.
For identical leaders we have A374517, ranks A374519.
For distinct leaders we have A374518, ranks A374638.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A373949.
- For leaders of weakly increasing runs we have A374637.
- For leaders of strictly increasing runs we have A374700.
- For leaders of weakly decreasing runs we have A374748.
- For leaders of strictly decreasing runs we have A374766.
A003242 counts anti-run compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
Cf. A124766, A238343, A261982, A333213, A374251, A374687, A374761.

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

A375140 Number of integer compositions of n whose leaders of weakly increasing runs are not strictly decreasing.

Original entry on oeis.org

0, 0, 0, 1, 3, 10, 26, 65, 151, 343, 750, 1614, 3410, 7123, 14724, 30220, 61639, 125166, 253233, 510936, 1028659, 2067620, 4150699, 8324552, 16683501, 33417933, 66910805, 133931495, 268023257, 536279457, 1072895973, 2146277961, 4293254010, 8587507415

Offset: 1

Gus Wiseman, Aug 10 2024

nonn

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

Also the number of integer compositions of n matching the dashed patterns 1-32 or 1-21.

			The a(1) = 0 through a(6) = 10 compositions:
     .  .  .  (121)  (131)   (132)
                     (1121)  (141)
                     (1211)  (1131)
                             (1212)
                             (1221)
                             (1311)
                             (2121)
                             (11121)
                             (11211)
                             (12111)

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 A188920.
Leaders of weakly increasing runs are rows of A374629, sum A374630.
For weakly decreasing leaders we have A374636, ranks A375137 or A375138.
For leaders of weakly decreasing runs we have the complement of A374746.
Compositions of this type are ranked by A375295, reverse A375296.
A003242 counts anti-runs, ranks A333489.
A106356 counts compositions by number of maximal anti-runs.
A238424 counts partitions whose first differences are an anti-run.
A274174 counts contiguous compositions, ranks A374249.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
A335548 counts non-contiguous compositions, ranks A374253.
A374637 counts compositions by sum of leaders of weakly increasing runs.
Cf. A124766, A238343, A261982, A333213, A335514, A373949, A374635, A374678, A374687, A374761.

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

a(n) = 2^(n-1) - A188920(n).

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

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Aug 02 2024

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

Are the column-sums finite?

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

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

Column n = k is A000041.
Row-sums are A011782.
For length instead of sum we have A333213.
The corresponding rank statistic is A374758, row-sums of A374757.
For identical leaders we have A374760, ranks A374759.
For distinct leaders we have A374761, ranks A374767.
Other types of runs (instead of strictly decreasing):
- For leaders of identical runs we have A373949.
- For leaders of anti-runs we have A374521.
- For leaders of weakly increasing runs we have A374637.
- For leaders of strictly increasing runs we have A374700.
- For leaders of weakly decreasing runs we have A374748.
A003242 counts anti-run compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
Cf. A106356, A238343, A261982, A274174, A374517, A374518, A374687.

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

A375139 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are not weakly decreasing.

Original entry on oeis.org

26, 50, 53, 58, 90, 98, 100, 101, 106, 107, 114, 117, 122, 154, 164, 178, 181, 186, 194, 196, 197, 201, 202, 203, 210, 212, 213, 214, 215, 218, 226, 228, 229, 234, 235, 242, 245, 250, 282, 306, 309, 314, 324, 329, 346, 354, 356, 357, 362, 363, 370, 373, 378

Offset: 1

Gus Wiseman, Aug 12 2024

nonn

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

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with corresponding compositions begin:
   26: (1,2,2)
   50: (1,3,2)
   53: (1,2,2,1)
   58: (1,1,2,2)
   90: (2,1,2,2)
   98: (1,4,2)
  100: (1,3,3)
  101: (1,3,2,1)
  106: (1,2,2,2)
  107: (1,2,2,1,1)
  114: (1,1,3,2)
  117: (1,1,2,2,1)
  122: (1,1,1,2,2)
  154: (3,1,2,2)
  164: (2,3,3)
  178: (2,1,3,2)
  181: (2,1,2,2,1)
  186: (2,1,1,2,2)

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

For leaders of identical runs we have A335485.
Ranked by positions of non-weakly decreasing rows in A374683.
For identical leaders we have A374685, counted by A374686.
The complement is counted by A374697.
For distinct leaders we have A374698, counted by A374687.
Compositions of this type are counted by A375135.
Weakly increasing leaders: A375137, counts A374636, complement A189076.
Interchanging weak/strict: A375295, counted by A375140, complement A188920.
A003242 counts anti-run compositions, ranks A333489.
A374700 counts compositions by sum of leaders of strictly increasing runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Leader is A065120.
- Parts are listed by A066099.
- Strict compositions are A233564.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Run-count: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Run-rank: A375123, A375124, A375125, A375126, A375127.
Cf. A000009, A261982, A333213, A374520, A374630, A374699, A374768.

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

cp's OEIS Frontend

A374697 Number of integer compositions of n whose leaders of strictly increasing 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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A374763 Number of integer compositions of n whose leaders of strictly decreasing runs are themselves strictly decreasing.

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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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A374521 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of anti-runs sum to k.

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A375140 Number of integer compositions of n whose leaders of weakly increasing runs are not strictly decreasing.

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