OFFSET
0,4
COMMENTS
First differs from A325534 at a(10) = 25, A325534(10) = 26. The first separable partition without an alternating permutation is (3,2,2,2,1).
A sequence is alternating 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 alternating 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(1) = 1 through a(8) = 14 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(21) (31) (32) (42) (43) (53)
(211) (41) (51) (52) (62)
(221) (321) (61) (71)
(311) (411) (322) (332)
(2211) (331) (422)
(421) (431)
(511) (521)
(3211) (611)
(22111) (3221)
(3311)
(4211)
(22211)
(32111)
MATHEMATICA
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[IntegerPartitions[n], Select[Permutations[#], wigQ]!={}&]], {n, 0, 15}]
CROSSREFS
Includes all strict partitions A000009.
Including twins (x,x) gives A344740.
The Heinz numbers of these partitions are A345172.
The version for factorizations is A348379.
A000041 counts integer partitions.
A001250 counts alternating permutations.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A344604 counts alternating compositions with twins.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 13 2021
EXTENSIONS
a(26)-a(32) from Robert Price, Jun 23 2021
a(33)-a(48) from Alois P. Heinz, Jun 23 2021
a(49) onwards from Joseph Likar, Sep 05 2023
STATUS
approved