

A345170


Number of integer partitions of n with an alternating permutation.


58



1, 1, 1, 2, 3, 5, 6, 10, 14, 19, 25, 36, 48, 64, 84, 111, 146, 191, 244, 315, 404, 515, 651, 823, 1035, 1295, 1616, 2011, 2492, 3076, 3787, 4650, 5695, 6952, 8463, 10280, 12460, 15059, 18162, 21858, 26254, 31463, 37641, 44933, 53554, 63704, 75653, 89683, 106162, 125445, 148020
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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 antirun permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2).


LINKS



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.
A001250 counts alternating permutations.
A003242 counts antirun compositions.
A344604 counts alternating compositions with twins.
Cf. A000070, A103919, A335126, A344605, A344653, A344654, A344742, A345164, A345166, A345167, A345168, A345195.


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



