A382771 Number of ways to permute the prime indices of n so that the run-lengths are all different.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 2, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 1, 2, 1, 2, 0, 2, 1, 2, 0, 2, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 2, 0, 0, 1, 2, 1, 0, 2, 2, 0, 0, 1, 2, 1, 0, 1, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Apr 07 2025

Keywords

• nonn

Comments

The first x with a(x) > 0 but A382857(x) > 1 is a(216) = 4, A382857(216) = 4.

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 a(96) = 4 permutations are:
  (1,1,1,1,1,2)
  (1,1,1,2,1,1)
  (1,1,2,1,1,1)
  (2,1,1,1,1,1)
The a(216) = 4 permutations are:
  (1,1,2,2,2,1)
  (1,2,2,2,1,1)
  (2,1,1,1,2,2)
  (2,2,1,1,1,2)
The a(360) = 6 permutations are:
  (1,1,1,2,2,3)
  (1,1,1,3,2,2)
  (2,2,1,1,1,3)
  (2,2,3,1,1,1)
  (3,1,1,1,2,2)
  (3,2,2,1,1,1)
		

Crossrefs

Positions of 1 are A000961.

Positions of positive terms are A351294, conjugate A381432.

Positions of 0 are A351295, conjugate A381433, equal A382879.

Sorted positions of first appearances are A382772, equal A382878.

For prescribed signature we have A382773, equal A382858.

The restriction to factorials is A382774, equal A335407.

For equal instead of distinct run-lengths we have A382857.

For run-sums instead of run-lengths we have A382876, equal A382877.

Positions of terms > 1 are A383113.

A044813 lists numbers whose binary expansion has distinct run-lengths.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A098859 counts partitions with distinct multiplicities, ordered A242882.

A239455 counts Look-and-Say partitions, complement A351293.

A329738 counts compositions with equal run-lengths, ranks A353744.

A329739 counts compositions with distinct run-lengths, ranks A351596.

Cf. A000720, A001221, A001222, A003242, A048767, A051903, A051904, A130091, A238130, A351013, A351202.

Programs

Mathematica

Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]],UnsameQ@@Length/@Split[#]&]],{n,30}]

Formula

a(A181821(n)) = a(A304660(n)) = A382773(n).

a(n!) = A382774(n).