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%I #12 Feb 03 2021 13:47:03
%S 1,1,1,0,1,2,1,0,0,2,1,1,1,2,2,0,1,1,1,1,2,2,1,0,0,2,0,1,1,6,1,0,2,2,
%T 2,2,1,2,2,0,1,6,1,1,1,2,1,0,0,1,2,1,1,0,2,0,2,2,1,6,1,2,1,0,2,6,1,1,
%U 2,6,1,1,1,2,1,1,2,6,1,0,0,2,1,6,2,2,2
%N Number of separations (Carlitz compositions or anti-runs) of the prime indices of n.
%C The first term that is not a factorial number is a(180) = 12.
%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 A separation (or Carlitz composition) of a multiset is a permutation with no adjacent equal parts.
%C a(n) depends only on the prime signature of n. - _Andrew Howroyd_, Feb 03 2021
%H Andrew Howroyd, <a href="/A335452/b335452.txt">Table of n, a(n) for n = 1..4096</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation_pattern">Permutation pattern</a>
%e The a(n) separations for n = 2, 6, 30, 180:
%e (1) (12) (123) (12123)
%e (21) (132) (12132)
%e (213) (12312)
%e (231) (12321)
%e (312) (13212)
%e (321) (21213)
%e (21231)
%e (21312)
%e (21321)
%e (23121)
%e (31212)
%e (32121)
%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_,x_,___}]&]],{n,100}]
%o (PARI)
%o F(i, j, r, t) = {sum(k=max(0, i-j), min(min(i,t), (i-j+t)\2), binomial(i, k)*binomial(r-i+1, t+i-j-2*k)*binomial(t-1, k-i+j))}
%o count(sig)={my(s=vecsum(sig), r=0, v=[1]); for(p=1, #sig, my(t=sig[p]); v=vector(s-r-t+1, j, sum(i=1, #v, v[i]*F(i-1, j-1, r, t))); r += t); v[1]}
%o a(n)={count(factor(n)[,2])} \\ _Andrew Howroyd_, Feb 03 2021
%Y Separations are counted by A003242 and ranked by A333489.
%Y Patterns are counted by A000670 and ranked by A333217.
%Y Permutations of prime indices are counted by A008480.
%Y Inseparable partitions are counted by A325535 and ranked by A335448.
%Y Cf. A000961, A005117, A056239, A112798, A181796, A261962, A333221, A335451, A335454, A335465, A335489.
%K nonn
%O 1,6
%A _Gus Wiseman_, Jun 21 2020
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