A374636 Number of integer compositions of n whose leaders of maximal weakly increasing runs are not weakly decreasing.

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

Author

Gus Wiseman, Aug 09 2024

Keywords

• nonn

Comments

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.

Examples

			- 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)
		

Links

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

Crossrefs

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.

Programs

Mathematica

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
				

Formula

a(n) = A011782(n) - A189076(n). - Jinyuan Wang, Feb 14 2025

Extensions

More terms from Jinyuan Wang, Feb 14 2025