A374679 Number of integer compositions of n whose leaders of anti-runs are strictly increasing.

1, 1, 1, 3, 4, 8, 15, 24, 45, 84, 142, 256, 464, 817, 1464, 2621, 4649, 8299, 14819, 26389, 47033, 83833, 149325, 266011, 473867, 843853

Offset

0,4

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.

Links

Table of n, a(n) for n=0..25.

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

Example

The a(0) = 1 through a(6) = 15 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)
                (12)  (13)   (14)   (15)
                (21)  (31)   (23)   (24)
                      (121)  (32)   (42)
                             (41)   (51)
                             (122)  (123)
                             (131)  (132)
                             (212)  (141)
                                    (213)
                                    (231)
                                    (312)
                                    (321)
                                    (1212)
                                    (1221)
                                    (2121)

Mathematica

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

Crossrefs

For distinct but not necessarily increasing leaders we have A374518.

For partitions instead of compositions we have A375134.

Other types of runs (instead of anti-):

- For leaders of identical runs we have A000041.

- For leaders of weakly increasing runs we have A374634.

- For leaders of strictly increasing runs we have A374688.

- For leaders of strictly decreasing runs we have A374762.

Other types of run-leaders (instead of strictly increasing):

- For identical leaders we have A374517.

- For distinct leaders we have A374518.

- For weakly increasing leaders we have A374681.

- 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. A188920, A238343, A333213, A333381, A373949, A374515, A374632, A374635, A374678, A374700, A374706, A375133.

Sequence in context: A104370 A310013 A033854 * A042981 A337438 A007486

Adjacent sequences: A374676 A374677 A374678 * A374680 A374681 A374682

Keyword

nonn,more

Author

Gus Wiseman, Aug 01 2024

Status

approved