%I #8 Jun 29 2020 22:21:19

%S 1,1,1,1,1,2,1,1,1,2,1,2,1,2,2,1,1,3,1,2,2,2,1,2,1,2,1,2,1,6,1,1,2,2,

%T 2,3,1,2,2,2,1,6,1,2,2,2,1,2,1,3,2,2,1,4,2,2,2,2,1,6,1,2,2,1,2,6,1,2,

%U 2,6,1,3,1,2,3,2,2,6,1,2,1,2,1,6,2,2,2

%N Number of (1,2,1)-avoiding 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 = 2, 10, 36, 54, 324, 30, 1458, 90:

%e (1) (13) (1122) (1222) (112222) (123) (1222222) (1223)

%e (31) (2112) (2122) (211222) (132) (2122222) (1322)

%e (2211) (2212) (221122) (213) (2212222) (2123)

%e (2221) (222112) (231) (2221222) (2213)

%e (222211) (312) (2222122) (2231)

%e (321) (2222212) (3122)

%e (2222221) (3212)

%e (3221)

%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 The matching version is A335446.

%Y Patterns are counted by A000670.

%Y (1,2,1)-avoiding patterns are counted by A001710.

%Y Permutations of prime indices are counted by A008480.

%Y Unsorted prime signature is A124010. Sorted prime signature is A118914.

%Y (1,2,1) and (2,1,2)-avoiding permutations of prime indices are counted by A333175.

%Y STC-numbers of permutations of prime indices are A333221.

%Y (1,2,1) and (2,1,2)-avoiding permutations of prime indices are A335448.

%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)-avoiding compositions are ranked by A335467.

%Y (1,2,1)-avoiding compositions are counted by A335471.

%Y Cf. A056239, A056986, A112798, A158005, A181796, A335452, A335463.

%K nonn

%O 1,6

%A _Gus Wiseman_, Jun 14 2020