proposed

approved

editing

proposed

Cf. `A189076, ~A228351, A238343, A333213, ~A333381, A373949, `A374515, A374632, `A374635, A374678, `A374700, ~A374706.

allocated for Gus WisemanNumber of integer compositions of n whose leaders of anti-runs are strictly decreasing.

1, 1, 1, 3, 5, 8, 16, 31, 52, 98, 179, 323, 590, 1078, 1945, 3531, 6421, 11621, 21041, 38116, 68904, 124562, 225138, 406513, 733710, 1323803

0,4

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.

Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.

The a(0) = 1 through a(6) = 16 compositions:

() (1) (2) (3) (4) (5) (6)

(12) (13) (14) (15)

(21) (31) (23) (24)

(121) (32) (42)

(211) (41) (51)

(131) (123)

(212) (132)

(311) (141)

(213)

(231)

(312)

(321)

(411)

(1212)

(2112)

(2121)

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

For distinct but not necessarily decreasing leaders we have A374518.

For partitions instead of compositions we have A375133.

Other types of runs (instead of anti-):

- For leaders of identical runs we have A000041.

- For leaders of weakly increasing runs we have A188920.

- For leaders of weakly decreasing runs we have A374746.

- For leaders of strictly decreasing runs we have A374763.

- For leaders of strictly increasing runs we have A374689.

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

- For identical leaders we have A374517, ranks A374519.

- For distinct leaders we have A374518, ranks A374638.

- For weakly increasing leaders we have A374681.

- For strictly increasing leaders we have A374679.

- For weakly decreasing leaders we have A374682.

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, ~A228351, A238343, A333213, ~A333381, A373949, `A374515, A374632, `A374635, A374678, `A374700, ~A374706.

allocated

nonn,more

Gus Wiseman, Aug 01 2024

approved

editing

allocated for Gus Wiseman

allocated

approved