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A335459 Number of permutations of the prime indices of n! with at least one non-singleton run.

%I A335459 #14 Apr 17 2021 18:13:48
%S A335459 0,0,0,0,4,18,102,786,3960,51450,675570,10804710,139674024,2793377664,
%T A335459 58662908640,1798893694080,26985313555200,782574083010720,
%U A335459 25992638958686400,857757034323189000,30021498596590300800,1563341714743040232000,64179292280096037844800,2631350957341279888915200
%N A335459 Number of permutations of the prime indices of n! with at least one non-singleton run.
%C A335459 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.
%H A335459 Andrew Howroyd, <a href="/A335459/b335459.txt">Table of n, a(n) for n = 0..200</a>
%F A335459 A008480(n!) = a(n) + A335407(n).
%e A335459 The a(4) = 4 and a(5) = 18 permutations:
%e A335459   (1,1,1,2)  (1,1,1,2,3)
%e A335459   (1,1,2,1)  (1,1,1,3,2)
%e A335459   (1,2,1,1)  (1,1,2,1,3)
%e A335459   (2,1,1,1)  (1,1,2,3,1)
%e A335459              (1,1,3,1,2)
%e A335459              (1,1,3,2,1)
%e A335459              (1,2,1,1,3)
%e A335459              (1,2,3,1,1)
%e A335459              (1,3,1,1,2)
%e A335459              (1,3,2,1,1)
%e A335459              (2,1,1,1,3)
%e A335459              (2,1,1,3,1)
%e A335459              (2,1,3,1,1)
%e A335459              (2,3,1,1,1)
%e A335459              (3,1,1,1,2)
%e A335459              (3,1,1,2,1)
%e A335459              (3,1,2,1,1)
%e A335459              (3,2,1,1,1)
%t A335459 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A335459 Table[Length[Select[Permutations[primeMS[n!]],MatchQ[#,{___,x_,x_,___}]&]],{n,0,10}]
%o A335459 (PARI) \\ See A335452 for count.
%o A335459 a(n)={my(sig=factor(n!)[, 2]); vecsum(sig)!/vecprod([k! | k<-sig]) - count(sig)} \\ _Andrew Howroyd_, Apr 17 2021
%Y A335459 The anti-run version is A335407.
%Y A335459 Anti-runs are ranked by A333489.
%Y A335459 Anti-run compositions are A003242.
%Y A335459 Anti-run patterns are A005649.
%Y A335459 Permutations of prime indices are A008480.
%Y A335459 Permutations of prime indices of n! are A325617.
%Y A335459 Anti-run permutations of prime indices are A335452.
%Y A335459 Cf. A001222, A022559, A056239, A106351, A112798, A114938, A325535, A335125.
%Y A335459 Factorial numbers: A000142, A002982, A007489, A027423, A054991, A108731, A325272, A325273, A325617.
%K A335459 nonn
%O A335459 0,5
%A A335459 _Gus Wiseman_, Jul 03 2020
%E A335459 a(11)-a(13) from _Vaclav Kotesovec_, Jul 07 2020
%E A335459 Terms a(14) and beyond from _Andrew Howroyd_, Apr 17 2021