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A335460 Number of (1,2,1) or (2,1,2)-matching permutations of the prime indices of n.

%I A335460 #11 Jun 29 2020 17:11:23
%S A335460 0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,2,0,0,0,1,0,0,0,0,0,0,
%T A335460 0,4,0,0,0,2,0,0,0,1,1,0,0,3,0,1,0,1,0,2,0,2,0,0,0,6,0,0,1,0,0,0,0,1,
%U A335460 0,0,0,8,0,0,1,1,0,0,0,3,0,0,0,6,0,0,0
%N A335460 Number of (1,2,1) or (2,1,2)-matching permutations of the prime indices of n.
%C A335460 Depends only on sorted prime signature (A118914).
%C A335460 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 A335460 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 A335460 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%H A335460 Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>
%e A335460 The a(n) compositions for n = 12, 24, 48, 36, 60, 72:
%e A335460   (121)  (1121)  (11121)  (1212)  (1213)  (11212)
%e A335460          (1211)  (11211)  (1221)  (1231)  (11221)
%e A335460                  (12111)  (2112)  (1312)  (12112)
%e A335460                           (2121)  (1321)  (12121)
%e A335460                                   (2131)  (12211)
%e A335460                                   (3121)  (21112)
%e A335460                                           (21121)
%e A335460                                           (21211)
%t A335460 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A335460 Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{___,x_,___,y_,___,x_,___}/;x!=y]&]],{n,100}]
%Y A335460 Positions of zeros are A303554.
%Y A335460 The (1,2,1)-matching part is A335446.
%Y A335460 The (2,1,2)-matching part is A335453.
%Y A335460 Replacing "or" with "and" gives A335462.
%Y A335460 Permutations of prime indices are counted by A008480.
%Y A335460 Unsorted prime signature is A124010. Sorted prime signature is A118914.
%Y A335460 STC-numbers of permutations of prime indices are A333221.
%Y A335460 (1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175.
%Y A335460 Patterns matched by standard compositions are counted by A335454.
%Y A335460 (1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
%Y A335460 Dimensions of downsets of standard compositions are A335465.
%Y A335460 Cf. A056239, A056986, A112798, A158005, A181796, A335451, A335452, A335463.
%K A335460 nonn
%O A335460 1,24
%A A335460 _Gus Wiseman_, Jun 20 2020