A374682 Number of integer compositions of n whose leaders of anti-runs are weakly decreasing.

1, 1, 2, 4, 8, 15, 30, 59, 114, 222, 434, 844, 1641, 3189, 6192, 12020, 23320, 45213, 87624, 169744, 328684, 636221, 1231067, 2381269, 4604713, 8901664

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) = 15 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (131)
                        (1111)  (212)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)
		

Links

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

Crossrefs

For reversed partitions instead of compositions we have A115029.

The complement is A374699.

Other types of runs (instead of anti-):

- For leaders of identical runs we have A000041.

- For leaders of weakly increasing runs we have A189076, complement A374636.

- For leaders of weakly decreasing runs we have A374747.

- For leaders of strictly decreasing runs we have A374765.

- For leaders of strictly increasing runs we have A374697.

Other types of run-leaders (instead of weakly 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 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. A238343, A333213, A333381, A373949, A374515, A374632, A374635, A374678, A374700, A374706.

Programs

Mathematica

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