A335407 Number of anti-run permutations of the prime indices of n!.

1, 1, 1, 2, 0, 2, 3, 54, 0, 30, 105, 6090, 1512, 133056, 816480, 127209600, 0, 10090080, 562161600, 69864795000, 49989139200, 29593652088000, 382147120555200, 41810689605484800, 4359985823793600, 3025062801079038720, 49052072750637116160, 25835971971637227375360

Offset: 0

Author

Gus Wiseman, Jul 01 2020

Keywords

• nonn

Comments

An anti-run is a sequence with no adjacent equal parts.

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.

Conjecture: Only vanishes at n = 4 and n = 8.

a(16) = 0. Proof: 16! = 2^15 * m where bigomega(m) = A001222(m) = 13. We can't separate 15 1's with 13 other numbers. - David A. Corneth, Jul 04 2020

Examples

			The a(0) = 1 through a(6) = 3 anti-run permutations:
  ()  ()  (1)  (1,2)  .  (1,2,1,3,1)  (1,2,1,2,1,3,1)
               (2,1)     (1,3,1,2,1)  (1,2,1,3,1,2,1)
                                      (1,3,1,2,1,2,1)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

The version for Mersenne numbers is A335432.

Anti-run compositions are A003242.

Anti-run patterns are counted by A005649.

Permutations of prime indices are A008480.

Anti-runs are ranked by A333489.

Separable partitions are ranked by A335433.

Inseparable partitions are ranked by A335448.

Anti-run permutations of prime indices are A335452.

Strict permutations of prime indices are A335489.

Cf. A056239, A106351, A112798, A114938, A278990, A292884, A325535, A335125.

Factorial numbers: A000142, A001222, A002982, A007489, A022559, A027423, A054991, A108731, A181819, A181821, A325272, A325273, A325617.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n!]],!MatchQ[#,{_,x_,x_,_}]&]],{n,0,10}]

PARI

\\ See A335452 for count.
a(n)={count(factor(n!)[,2])} \\ Andrew Howroyd, Feb 03 2021

Formula

a(n) = A335452(A000142(n)). - Andrew Howroyd, Feb 03 2021

Extensions

Terms a(14) and beyond from Andrew Howroyd, Feb 03 2021