A382858 Number of ways to permute a multiset whose multiplicities are the prime indices of n so that the run-lengths are all equal.

1, 1, 1, 2, 1, 1, 1, 6, 4, 0, 1, 6, 1, 0, 1, 24, 1, 12, 1, 2, 1, 0, 1, 36, 4, 0, 36, 0, 1, 10, 1, 120, 0, 0, 1, 84, 1, 0, 0, 24, 1, 3, 1, 0, 38, 0, 1, 240, 6, 18, 0, 0, 1, 246, 0, 6, 0, 0, 1, 96, 1, 0, 30, 720, 1, 0, 1, 0, 0, 14, 1, 660, 1, 0, 74, 0, 1, 0, 1

Offset: 1

Author

Gus Wiseman, Apr 09 2025

Keywords

• nonn

Comments

This described multiset (row n of A305936, Heinz number A181821) is generally not the same as the multiset of prime indices of n (A112798). For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Examples

			The a(9) = 4 permutations are:
  (1,1,2,2)
  (1,2,1,2)
  (2,1,2,1)
  (2,2,1,1)
		

Crossrefs

The anti-run case is A335125.

These permutations for factorials are counted by A335407, distinct A382774.

For distinct instead of equal run-lengths we have A382773.

For prime indices we have A382857 (firsts A382878), distinct A382771 (firsts A382772).

Positions of 0 are A382914, signature restriction of A382915.

A003963 gives product of prime indices.

A140690 lists numbers whose binary expansion has equal run-lengths, distinct A044813.

A047966 counts partitions with equal multiplicities, distinct A098859.

A056239 adds up prime indices, row sums of A112798.

A304442 counts partitions with equal run-sums, ranks A353833.

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

A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.

A382913 ranks Look-and-Say partitions by signature, complement A382912.

Cf. A000720, A000961, A001221, A001222, A003242, A048767, A181821, A182854, A238130, A305936, A351202, A382879.

Programs

Mathematica

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Table[Length[Select[Permutations[nrmptn[n]],SameQ@@Length/@Split[#]&]],{n,100}]

Formula

a(n) = A382857(A181821(n)) = A382857(A304660(n)).