A374699 Number of integer compositions of n whose leaders of maximal anti-runs are not weakly decreasing.

0, 0, 0, 0, 0, 1, 2, 5, 14, 34, 78, 180, 407, 907, 2000, 4364, 9448, 20323, 43448, 92400, 195604, 412355, 866085, 1813035, 3783895, 7875552

Offset

0,7

Comments

The leaders of maximal 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) = 0 through a(8) = 14 compositions:
  .  .  .  .  .  (122)  (1122)  (133)    (233)
                        (1221)  (1222)   (1133)
                                (11122)  (1223)
                                (11221)  (1322)
                                (12211)  (1331)
                                         (11222)
                                         (12122)
                                         (12212)
                                         (12221)
                                         (21122)
                                         (111122)
                                         (111221)
                                         (112211)
                                         (122111)

Mathematica

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

Crossrefs

The complement is counted by A374682.

Other types of runs (instead of anti-):

- For leaders of identical runs we have A056823.

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

- For leaders of strictly increasing runs: A375135, complement A374697.

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

- For identical leaders we have A374640, ranks A374520, complement A374517, ranks A374519.

- For distinct leaders we have A374678, ranks A374639, complement A374518, ranks A374638.

- For weakly increasing leaders we have complement A374681.

- For strictly increasing leaders we have complement complement A374679.

- For strictly decreasing leaders we have complement 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.

A333381 counts maximal anti-runs in standard compositions.

Cf. A115029, A238343, A333213, A373949, A374515, A374632, A374635, A374700.

Sequence in context: A018015 A080039 A344236 * A265226 A357835 A369591

Adjacent sequences: A374696 A374697 A374698 * A374700 A374701 A374704

Keyword

nonn,more,new

Author

Gus Wiseman, Aug 06 2024

Status

approved