A335407 -id:A335407 - cp's OEIS frontend

A382774 Number of ways to permute the prime indices of n! so that the run-lengths are all different.

1, 1, 1, 0, 2, 0, 6, 0, 0, 0, 96, 0

Offset: 0

Gus Wiseman, Apr 09 2025

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, sum A056239.

			The prime indices of 24 are {1,1,1,2}, with permutations (1,1,1,2) and (2,1,1,1), so a(4) = 2.

For anti-run permutations we have A335407, see also A335125, A382858.
This is the restriction of A382771 to the factorials A000142, equal A382857.
A022559 counts prime indices of n!, sum A081401.
A044813 lists numbers whose binary expansion has distinct run-lengths, equal A140690.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A328592 lists numbers whose binary form has distinct runs of ones, equal A164707.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000720, A001221, A001222, A003242, A048767, A130091, A181821, A351013, A351202, A360015, A382773, A382879.

Table[Length[Select[Permutations[prix[n!]],UnsameQ@@Length/@Split[#]&]],{n,0,6}]

a(n) = A382771(n!).

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

Original entry on oeis.org

0, 0, 0, 0, 4, 18, 102, 786, 3960, 51450, 675570, 10804710, 139674024, 2793377664, 58662908640, 1798893694080, 26985313555200, 782574083010720, 25992638958686400, 857757034323189000, 30021498596590300800, 1563341714743040232000, 64179292280096037844800, 2631350957341279888915200

Offset: 0

Gus Wiseman, Jul 03 2020

nonn

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.

			The a(4) = 4 and a(5) = 18 permutations:
  (1,1,1,2)  (1,1,1,2,3)
  (1,1,2,1)  (1,1,1,3,2)
  (1,2,1,1)  (1,1,2,1,3)
  (2,1,1,1)  (1,1,2,3,1)
             (1,1,3,1,2)
             (1,1,3,2,1)
             (1,2,1,1,3)
             (1,2,3,1,1)
             (1,3,1,1,2)
             (1,3,2,1,1)
             (2,1,1,1,3)
             (2,1,1,3,1)
             (2,1,3,1,1)
             (2,3,1,1,1)
             (3,1,1,1,2)
             (3,1,1,2,1)
             (3,1,2,1,1)
             (3,2,1,1,1)

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

The anti-run version is A335407.
Anti-runs are ranked by A333489.
Anti-run compositions are A003242.
Anti-run patterns are A005649.
Permutations of prime indices are A008480.
Permutations of prime indices of n! are A325617.
Anti-run permutations of prime indices are A335452.
Cf. A001222, A022559, A056239, A106351, A112798, A114938, A325535, A335125.
Factorial numbers: A000142, A002982, A007489, A027423, A054991, A108731, A325272, A325273, A325617.

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}]

\\ See A335452 for count.
a(n)={my(sig=factor(n!)[, 2]); vecsum(sig)!/vecprod([k! | k<-sig]) - count(sig)} \\ Andrew Howroyd, Apr 17 2021

A008480(n!) = a(n) + A335407(n).

a(11)-a(13) from Vaclav Kotesovec, Jul 07 2020

Terms a(14) and beyond from Andrew Howroyd, Apr 17 2021

cp's OEIS Frontend

A382774 Number of ways to permute the prime indices of n! so that the run-lengths are all different.

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