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%I #16 Jun 29 2020 22:20:46
%S 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,
%T 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,
%U 0,0,0,7,0,0,0,1,0,0,0,3,0,0,0,6,0,0,0
%N Number of (1,2,1)-matching permutations of the prime indices of n.
%C Depends only on unsorted prime signature (A124010), but not only on sorted prime signature (A118914).
%C 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.
%C 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).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H Gus Wiseman, <a href="/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%e The a(n) permutations for n = 12, 24, 36, 60, 72, 90, 120, 144:
%e (121) (1121) (1212) (1213) (11212) (1232) (11213) (111212)
%e (1211) (1221) (1231) (11221) (2132) (11231) (111221)
%e (2121) (1312) (12112) (2312) (11312) (112112)
%e (1321) (12121) (2321) (11321) (112121)
%e (2131) (12211) (12113) (112211)
%e (3121) (21121) (12131) (121112)
%e (21211) (12311) (121121)
%e (13112) (121211)
%e (13121) (122111)
%e (13211) (211121)
%e (21131) (211211)
%e (21311) (212111)
%e (31121)
%e (31211)
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{___,x_,___,y_,___,x_,___}/;x<y]&]],{n,100}]
%Y Positions of zeros are A065200.
%Y The avoiding version is A335449.
%Y Patterns are counted by A000670.
%Y Permutations of prime indices are counted by A008480.
%Y Unimodal permutations of prime indices are counted by A332288.
%Y (1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175.
%Y STC-numbers of permutations of prime indices are A333221.
%Y Patterns matched by standard compositions are counted by A335454.
%Y (1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
%Y (1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
%Y Dimensions of downsets of standard compositions are A335465.
%Y (1,2,1)-matching compositions are ranked by A335466.
%Y (1,2,1)-matching compositions are counted by A335470.
%Y (1,2,1)-matching patterns are counted by A335509.
%Y Cf. A056239, A056986, A112798, A158005, A158009, A181796, A335452, A335463.
%K nonn
%O 1,24
%A _Gus Wiseman_, Jun 13 2020
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