A374518 Number of integer compositions of n whose leaders of anti-runs are distinct.

1, 1, 1, 3, 5, 9, 17, 32, 58, 112, 201, 371, 694, 1276, 2342, 4330, 7958, 14613, 26866, 49303, 90369, 165646, 303342, 555056, 1015069, 1855230

Offset: 0

Author

Gus Wiseman, Aug 01 2024

Keywords

• nonn

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(6) = 17 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)
                (12)  (13)   (14)   (15)
                (21)  (31)   (23)   (24)
                      (121)  (32)   (42)
                      (211)  (41)   (51)
                             (122)  (123)
                             (131)  (132)
                             (212)  (141)
                             (311)  (213)
                                    (231)
                                    (312)
                                    (321)
                                    (411)
                                    (1212)
                                    (1221)
                                    (2112)
                                    (2121)
		

Links

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

Crossrefs

These compositions have ranks A374638.

The complement is counted by A374678.

For partitions instead of compositions we have A375133.

Other types of runs (instead of anti-):

- For leaders of identical runs we have A274174, ranks A374249.

- For leaders of weakly increasing runs we have A374632, ranks A374768.

- For leaders of strictly increasing runs we have A374687, ranks A374698.

- For leaders of weakly decreasing runs we have A374743, ranks A374701.

- For leaders of strictly decreasing runs we have A374761, ranks A374767.

Other types of run-leaders (instead of distinct):

- For identical leaders we have A374517.

- For weakly increasing leaders we have A374681.

- 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.

Cf. A188920, A233564, A238343, A333213, A333381, A373949, A374515.

Programs

Mathematica

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