cp's OEIS Frontend

A335452 Number of separations (Carlitz compositions or anti-runs) of the prime indices of n.

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