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A335521
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Number of (1,2,3)-avoiding permutations of the prime indices of n.
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5
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1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 1, 2, 2, 2, 6, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 10, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2
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OFFSET
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1,6
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COMMENTS
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A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).
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LINKS
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FORMULA
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EXAMPLE
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The a(n) permutations for n = 1, 6, 12, 24, 30, 36, 60, 72, 120:
() (12) (112) (1112) (132) (1122) (1132) (11122) (11132)
(21) (121) (1121) (213) (1212) (1312) (11212) (11312)
(211) (1211) (231) (1221) (1321) (11221) (11321)
(2111) (312) (2112) (2113) (12112) (13112)
(321) (2121) (2131) (12121) (13121)
(2211) (2311) (12211) (13211)
(3112) (21112) (21113)
(3121) (21121) (21131)
(3211) (21211) (21311)
(22111) (23111)
(31112)
(31121)
(31211)
(32111)
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Length[Select[Permutations[primeMS[n]], !MatchQ[#, {___, x_, ___, y_, ___, z_, ___}/; x<y<z]&]], {n, 100}]
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CROSSREFS
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These compositions are counted by A102726.
Patterns avoiding this pattern are counted by A226316.
The complement A335520 is the matching version.
Permutations of prime indices are counted by A008480.
Anti-run permutations of prime indices are counted by A335452.
Cf. A056239, A056986, A112798, A238279, A281188, A333221, A333755, A335456, A335460, A335462, A335463.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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