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A344614
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Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x < y < z or x > y > z.
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50
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1, 1, 2, 4, 8, 16, 30, 58, 110, 209, 397, 753, 1429, 2711, 5143, 9757, 18511, 35117, 66621, 126389, 239781, 454897, 863010, 1637260, 3106138, 5892821, 11179603, 21209446, 40237641, 76337091, 144823431, 274752731, 521249018, 988891100, 1876081530, 3559220898, 6752400377
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OFFSET
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0,3
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COMMENTS
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These compositions avoid the strict consecutive patterns (1,2,3) and (3,2,1), the weak version being A344604.
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LINKS
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EXAMPLE
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The a(6) = 30 compositions are:
(6) (15) (114) (1113) (11112) (111111)
(24) (132) (1122) (11121)
(33) (141) (1131) (11211)
(42) (213) (1212) (12111)
(51) (222) (1221) (21111)
(231) (1311)
(312) (2112)
(411) (2121)
(2211)
(3111)
Missing are: (123), (321).
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MATHEMATICA
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !MatchQ[#, {___, x_, y_, z_, ___}/; x<y<z||x>y>z]&]], {n, 0, 15}]
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CROSSREFS
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A001250 counts alternating permutations.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A325534 counts separable partitions.
A325535 counts inseparable partitions.
A344604 counts wiggly compositions with twins.
A344605 counts wiggly patterns with twins.
A344606 counts wiggly permutations of prime factors with twins.
Counting compositions by patterns:
- A106351 avoiding (1,1) adjacent by sum and length.
- A128695 avoiding (1,1,1) adjacent.
- A344604 weakly avoiding (1,2,3) and (3,2,1) adjacent.
- A344614 avoiding (1,2,3) and (3,2,1) adjacent.
- A344615 weakly avoiding (1,2,3) adjacent.
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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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