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A335446
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Number of (1,2,1)-matching permutations of the prime indices of n.
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15
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 7, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 6, 0, 0, 0
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OFFSET
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1,24
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COMMENTS
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Depends only on unsorted prime signature (A124010), but not only on sorted prime signature (A118914).
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. 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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EXAMPLE
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The a(n) permutations for n = 12, 24, 36, 60, 72, 90, 120, 144:
(121) (1121) (1212) (1213) (11212) (1232) (11213) (111212)
(1211) (1221) (1231) (11221) (2132) (11231) (111221)
(2121) (1312) (12112) (2312) (11312) (112112)
(1321) (12121) (2321) (11321) (112121)
(2131) (12211) (12113) (112211)
(3121) (21121) (12131) (121112)
(21211) (12311) (121121)
(13112) (121211)
(13121) (122111)
(13211) (211121)
(21131) (211211)
(21311) (212111)
(31121)
(31211)
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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_, ___, x_, ___}/; x<y]&]], {n, 100}]
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CROSSREFS
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Permutations of prime indices are counted by A008480.
Unimodal permutations of prime indices are counted by A332288.
(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175.
STC-numbers of permutations of prime indices are A333221.
Patterns matched by standard compositions are counted by A335454.
(1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
Dimensions of downsets of standard compositions are A335465.
(1,2,1)-matching compositions are ranked by A335466.
(1,2,1)-matching compositions are counted by A335470.
(1,2,1)-matching patterns are counted by A335509.
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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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