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A335462 Number of (1,2,1) and (2,1,2)-matching permutations of the prime indices of n.

%I A335462 #10 Jun 29 2020 17:10:34
%S A335462 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%T A335462 0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A335462 0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A335462 Number of (1,2,1) and (2,1,2)-matching permutations of the prime indices of n.
%C A335462 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 A335462 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).
%H A335462 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A335462 Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%e A335462 The a(n) permutations for n = 36, 72, 270, 144, 300:
%e A335462   (1,2,1,2)  (1,1,2,1,2)  (2,1,2,3,2)  (1,1,1,2,1,2)  (1,2,3,1,3)
%e A335462   (2,1,2,1)  (1,2,1,1,2)  (2,1,3,2,2)  (1,1,2,1,1,2)  (1,3,1,2,3)
%e A335462              (1,2,1,2,1)  (2,2,1,3,2)  (1,1,2,1,2,1)  (1,3,1,3,2)
%e A335462              (2,1,1,2,1)  (2,2,3,1,2)  (1,2,1,1,1,2)  (1,3,2,1,3)
%e A335462              (2,1,2,1,1)  (2,3,1,2,2)  (1,2,1,1,2,1)  (1,3,2,3,1)
%e A335462                           (2,3,2,1,2)  (1,2,1,2,1,1)  (2,1,3,1,3)
%e A335462                                        (2,1,1,1,2,1)  (2,3,1,3,1)
%e A335462                                        (2,1,1,2,1,1)  (3,1,2,1,3)
%e A335462                                        (2,1,2,1,1,1)  (3,1,2,3,1)
%e A335462                                                       (3,1,3,1,2)
%e A335462                                                       (3,1,3,2,1)
%e A335462                                                       (3,2,1,3,1)
%t A335462 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A335462 Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{___,x_,___,y_,___,x_,___}/;x<y]&&MatchQ[#,{___,x_,___,y_,___,x_,___}/;x>y]&]],{n,100}]
%Y A335462 The avoiding version is A333175.
%Y A335462 Replacing "and" with "or" gives A335460.
%Y A335462 Positions of nonzero terms are A335463.
%Y A335462 Permutations of prime indices are counted by A008480.
%Y A335462 Unsorted prime signature is A124010. Sorted prime signature is A118914.
%Y A335462 STC-numbers of permutations of prime indices are A333221.
%Y A335462 Patterns matched by standard compositions are counted by A335454.
%Y A335462 Dimensions of downsets of standard compositions are A335465.
%Y A335462 Cf. A056239, A056986, A112798, A158005, A181796, A335451, A335452.
%K A335462 nonn
%O A335462 1,36
%A A335462 _Gus Wiseman_, Jun 20 2020