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A344654
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Number of integer partitions of n of which every permutation has a consecutive monotone triple, i.e., a triple (..., x, y, z, ...) such that either x <= y <= z or x >= y >= z.
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53
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0, 0, 0, 1, 1, 2, 4, 5, 7, 11, 16, 20, 28, 37, 50, 65, 84, 106, 140, 175, 222, 277, 350, 432, 539, 663, 819, 999, 1225, 1489, 1816, 2192, 2653, 3191, 3846, 4603, 5516, 6578, 7852, 9327, 11083, 13120, 15532, 18328, 21620, 25430, 29904, 35071, 41110, 48080
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OFFSET
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0,6
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Select[Permutations[#], !MatchQ[#, {___, x_, y_, z_, ___}/; x<=y<=z||x>=y>=z]&]=={}&]], {n, 15}]
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CROSSREFS
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The Heinz numbers of these partitions are A344653, complement A344742.
The complement is counted by A344740.
The normal case starts 0, 0, 0, then becomes A345162, complement A345163.
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.
A345170 counts partitions with a wiggly permutation, ranked by A345172.
A345192 counts non-wiggly compositions.
Cf. A000041, A000070, A102726, A103919, A333489, A335126, A344607, A344615, A345166, A345168, A345169.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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