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Number of (1,2,1) or (2,1,2)-matching permutations of the prime indices of n.
22

%I #11 Jun 29 2020 17:11:23

%S 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 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 0,0,0,8,0,0,1,1,0,0,0,3,0,0,0,6,0,0,0

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

%C Depends 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 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>

%H Gus Wiseman, <a href="https://oeis.org/A102726/a102726.txt">Sequences counting and ranking compositions by the patterns they match or avoid.</a>

%e The a(n) compositions for n = 12, 24, 48, 36, 60, 72:

%e (121) (1121) (11121) (1212) (1213) (11212)

%e (1211) (11211) (1221) (1231) (11221)

%e (12111) (2112) (1312) (12112)

%e (2121) (1321) (12121)

%e (2131) (12211)

%e (3121) (21112)

%e (21121)

%e (21211)

%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 A303554.

%Y The (1,2,1)-matching part is A335446.

%Y The (2,1,2)-matching part is A335453.

%Y Replacing "or" with "and" gives A335462.

%Y Permutations of prime indices are counted by A008480.

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

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

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

%Y Patterns matched by standard compositions are counted by A335454.

%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 Cf. A056239, A056986, A112798, A158005, A181796, A335451, A335452, A335463.

%K nonn

%O 1,24

%A _Gus Wiseman_, Jun 20 2020