

A345165


Number of integer partitions of n without an alternating permutation.


50



0, 0, 1, 1, 2, 2, 5, 5, 8, 11, 17, 20, 29, 37, 51, 65, 85, 106, 141, 175, 223, 277, 351, 432, 540, 663
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OFFSET

0,5


COMMENTS

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it has the antirun permutations (2,3,2,1,2) and (2,1,2,3,2).


LINKS



EXAMPLE

The a(2) = 1 through a(9) = 11 partitions:
(11) (111) (22) (2111) (33) (2221) (44) (333)
(1111) (11111) (222) (4111) (2222) (3222)
(3111) (31111) (5111) (6111)
(21111) (211111) (41111) (22221)
(111111) (1111111) (221111) (51111)
(311111) (321111)
(2111111) (411111)
(11111111) (2211111)
(3111111)
(21111111)
(111111111)


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

The Heinz numbers of these partitions are A345171.
A003242 counts antirun compositions.
A025047 counts alternating or wiggly compositions.
A344604 counts alternating compositions with twins.
A345164 counts alternating permutations of prime indices, w/ twins A344606.
Cf. A000070, A025048, A025049, A103919, A335126, A344605, A344607, A344615, A344653, A345166, A345167, A345168, A345169, A347706, A348609.


KEYWORD

nonn,more


AUTHOR



STATUS

approved



