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

Author

Gus Wiseman, Apr 09 2025

Keywords

• nonn
• more

Comments

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.

Examples

			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.
		

Crossrefs

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.

Programs

Mathematica

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

Formula

a(n) = A382771(n!).