OFFSET
0,3
EXAMPLE
The a(1) = 1 through a(6) = 19 compositions:
(1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(21) (22) (23) (24)
(111) (31) (32) (33)
(121) (41) (42)
(211) (131) (51)
(1111) (212) (141)
(221) (222)
(311) (231)
(1211) (312)
(2111) (321)
(11111) (411)
(1212)
(1311)
(2121)
(2211)
(3111)
(12111)
(21111)
(111111)
For example, (2,3,1,2) is such a composition, because the non-adjacent pairs of parts are (2,1), (2,2), (3,2), all of which are weakly decreasing.
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !MatchQ[#, {___, x_, __, y_, ___}/; y>x]&]], {n, 0, 15}]
CROSSREFS
Unimodal compositions are A001523.
The case of normal sequences appears to be A028859.
A version for ordered set partitions is A332872.
The case of strict compositions is A333150.
The version for strictly decreasing parts is A333193.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, May 16 2020
STATUS
approved