A374632 -id:A374632 - cp's OEIS frontend

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

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}]

A375297 Number of integer compositions of n matching both of the dashed patterns 23-1 and 1-32.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 21, 68, 199, 545, 1410, 3530, 8557, 20255, 46968, 107135, 240927, 535379, 1177435, 2566618, 5551456

Offset: 0

Gus Wiseman, Aug 23 2024

Also the number of integer compositions of n whose leaders of maximal weakly increasing runs are not weakly decreasing and whose reverse satisfies the same condition.

			The a(0) = 0 through a(11) = 21 compositions:
  .  .  .  .  .  .  .  .  .  (12321)  (1342)    (1352)
                                      (2431)    (2531)
                                      (12421)   (11342)
                                      (13231)   (12431)
                                      (112321)  (12521)
                                      (123211)  (13241)
                                                (13421)
                                                (14231)
                                                (23132)
                                                (24311)
                                                (112421)
                                                (113231)
                                                (122321)
                                                (123212)
                                                (123221)
                                                (124211)
                                                (132311)
                                                (212321)
                                                (1112321)
                                                (1123211)
                                                (1232111)

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 A332834.
For just one of the two conditions we have A374636, ranks A375137/A375138.
These compositions are ranked by A375407.
A003242 counts anti-runs, ranks A333489.
A011782 counts compositions.
A106356 counts compositions by number of maximal anti-runs.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A335456 counts patterns matched by compositions.
Cf. A000041, A056823, A188920, A189076, A238343, A333213, A335514, A374631, A374632, A374635, A374681.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MatchQ[#,{_,y_,z_,_,x_,_}/;x_,x_,_,z_,y_,_}/;x

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A375297 Number of integer compositions of n matching both of the dashed patterns 23-1 and 1-32.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica