OFFSET
0,6
COMMENTS
Such a permutation is characterized by being neither a twin (x,x) nor wiggly (A025047, A345192). A sequence is wiggly if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,3,2,2,2,2,1) has no wiggly permutations, even though it has the anti-run permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2).
LINKS
Joseph Likar, Table of n, a(n) for n = 0..1000
EXAMPLE
The a(3) = 1 through a(9) = 11 partitions:
(111) (1111) (2111) (222) (2221) (2222) (333)
(11111) (3111) (4111) (5111) (3222)
(21111) (31111) (41111) (6111)
(111111) (211111) (221111) (22221)
(1111111) (311111) (51111)
(2111111) (321111)
(11111111) (411111)
(2211111)
(3111111)
(21111111)
(111111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Select[Permutations[#], !MatchQ[#, {___, x_, y_, z_, ___}/; x<=y<=z||x>=y>=z]&]=={}&]], {n, 15}]
CROSSREFS
The complement is counted by A344740.
A001250 counts wiggly permutations.
A003242 counts anti-run compositions.
A344604 counts wiggly compositions with twins.
A344605 counts wiggly patterns with twins.
A344606 counts wiggly permutations of prime indices with twins.
A344614 counts compositions with no consecutive strictly monotone triple.
A345164 counts wiggly permutations of prime indices.
A345192 counts non-wiggly compositions.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 12 2021
EXTENSIONS
a(26)-a(32) from Robert Price, Jun 22 2021
a(33) onwards from Joseph Likar, Sep 06 2023
STATUS
approved