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

Author

Gus Wiseman, Aug 01 2024

Keywords

• nonn
• more

Comments

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.

Examples

			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)
		

Links

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

Crossrefs

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.

Programs

Mathematica

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