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A345165 Number of integer partitions of n without an alternating permutation. 51
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, 820, 999, 1226, 1489, 1817, 2192, 2654, 3191, 3847, 4603, 5517, 6578, 7853, 9327, 11084, 13120, 15533, 18328, 21621, 25430, 29905, 35071, 41111, 48080, 56206, 65554, 76420, 88918 (list; graph; refs; listen; history; text; internal format)
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 anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).
LINKS
Joseph Likar, Java Implementation using QBinomials
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
Excluding twins (x,x) gives A344654, complement A344740.
The normal case is A345162, complement A345163.
The complement is counted by A345170, ranked by A345172.
The Heinz numbers of these partitions are A345171.
The version for factorizations is A348380, complement A348379.
A version for ordered factorizations is A348613, complement A348610.
A000041 counts integer partitions.
A001250 counts alternating permutations, complement A348615.
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A025047 counts alternating or wiggly compositions.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A344604 counts alternating compositions with twins.
A345164 counts alternating permutations of prime indices, w/ twins A344606.
A345192 counts non-alternating compositions, without twins A348377.
Sequence in context: A240495 A304393 A325535 * A062405 A368180 A071181
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 12 2021
EXTENSIONS
a(26) onwards by Joseph Likar, Aug 21 2023
STATUS
approved

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Last modified July 17 10:03 EDT 2024. Contains 374375 sequences. (Running on oeis4.)