cp's OEIS Frontend

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

%I A382858 #6 Apr 10 2025 23:22:30
%S A382858 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,
%T A382858 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,
%U A382858 1,0,30,720,1,0,1,0,0,14,1,660,1,0,74,0,1,0,1
%N A382858 Number of ways to permute a multiset whose multiplicities are the prime indices of n so that the run-lengths are all equal.
%C A382858 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}.
%F A382858 a(n) = A382857(A181821(n)) = A382857(A304660(n)).
%e A382858 The a(9) = 4 permutations are:
%e A382858   (1,1,2,2)
%e A382858   (1,2,1,2)
%e A382858   (2,1,2,1)
%e A382858   (2,2,1,1)
%t A382858 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A382858 Table[Length[Select[Permutations[nrmptn[n]],SameQ@@Length/@Split[#]&]],{n,100}]
%Y A382858 The anti-run case is A335125.
%Y A382858 These permutations for factorials are counted by A335407, distinct A382774.
%Y A382858 For distinct instead of equal run-lengths we have A382773.
%Y A382858 For prime indices we have A382857 (firsts A382878), distinct A382771 (firsts A382772).
%Y A382858 Positions of 0 are A382914, signature restriction of A382915.
%Y A382858 A003963 gives product of prime indices.
%Y A382858 A140690 lists numbers whose binary expansion has equal run-lengths, distinct A044813.
%Y A382858 A047966 counts partitions with equal multiplicities, distinct A098859.
%Y A382858 A056239 adds up prime indices, row sums of A112798.
%Y A382858 A304442 counts partitions with equal run-sums, ranks A353833.
%Y A382858 A329738 counts compositions with equal run-lengths, ranks A353744.
%Y A382858 A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
%Y A382858 A382913 ranks Look-and-Say partitions by signature, complement A382912.
%Y A382858 Cf. A000720, A000961, A001221, A001222, A003242, A048767, A181821, A182854, A238130, A305936, A351202, A382879.
%K A382858 nonn
%O A382858 1,4
%A A382858 _Gus Wiseman_, Apr 09 2025