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%I #11 Jun 29 2020 22:21:07
%S 0,0,0,0,0,1,0,0,0,1,0,2,0,1,1,0,0,2,0,2,1,1,0,3,0,1,0,2,0,5,0,0,1,1,
%T 1,5,0,1,1,3,0,5,0,2,2,1,0,4,0,2,1,2,0,3,1,3,1,1,0,11,0,1,2,0,1,5,0,2,
%U 1,5,0,9,0,1,2,2,1,5,0,4,0,1,0,11,1,1
%N Number of (1,2)-matching permutations of the prime indices of n.
%C Depends only on sorted prime signature (A118914).
%C Also the number of (2,1)-matching permutations of the prime indices of n.
%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>
%F a(n) = A008480(n) - 1.
%e The a(n) permutations for n = 6, 12, 24, 48, 30, 72, 60:
%e (12) (112) (1112) (11112) (123) (11122) (1123)
%e (121) (1121) (11121) (132) (11212) (1132)
%e (1211) (11211) (213) (11221) (1213)
%e (12111) (231) (12112) (1231)
%e (312) (12121) (1312)
%e (12211) (1321)
%e (21112) (2113)
%e (21121) (2131)
%e (21211) (2311)
%e (3112)
%e (3121)
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Table[Length[Select[Permutations[primeMS[n]],!GreaterEqual@@#&]],{n,100}]
%Y The avoiding version is A000012.
%Y Patterns are counted by A000670.
%Y Positions of zeros are A000961.
%Y (1,2)-matching patterns are counted by A002051.
%Y Permutations of prime indices are counted by A008480.
%Y (1,2)-matching compositions are counted by A056823.
%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)-matching compositions are ranked by A335485.
%Y Cf. A056239, A056986, A112798, A181796, A333175, A335451, A335452, A335463, A333175.
%K nonn
%O 1,12
%A _Gus Wiseman_, Jun 14 2020
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