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A335446
Number of (1,2,1)-matching permutations of the prime indices of n.
15
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
OFFSET
1,24
COMMENTS
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).
EXAMPLE
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)
MATHEMATICA
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}]
CROSSREFS
Positions of zeros are A065200.
The avoiding version is A335449.
Patterns are counted by A000670.
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.
Sequence in context: A294929 A076948 A255309 * A376514 A335460 A367077
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 13 2020
STATUS
approved