OFFSET

1,5

COMMENTS

The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.

Also the number of integer compositions of n matching the dashed patterns 1-32 or 1-21.

LINKS

FORMULA

a(n) = 2^(n-1) - A188920(n).

EXAMPLE

The a(1) = 0 through a(6) = 10 compositions:

. . . (121) (131) (132)

(1121) (141)

(1211) (1131)

(1212)

(1221)

(1311)

(2121)

(11121)

(11211)

(12111)

MATHEMATICA

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

- or -

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MatchQ[#, {___, x_, ___, z_, y_, ___}/; x<=y<z]&]], {n, 0, 15}]

CROSSREFS

For leaders of identical runs we have A056823.

The complement is counted by A188920.

For leaders of weakly decreasing runs we have the complement of A374746.

A106356 counts compositions by number of maximal anti-runs.

A238424 counts partitions whose first differences are an anti-run.

A011782 counts compositions.

A335456 counts patterns matched by compositions.

A374637 counts compositions by sum of leaders of weakly increasing runs.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 10 2024

STATUS

approved