cp's OEIS Frontend

A382773 Number of ways to permute a multiset whose multiplicities are the prime indices of n so that the run-lengths are all different.

%I A382773 #7 Apr 09 2025 23:39:26
%S A382773 1,1,1,0,1,2,1,0,0,2,1,0,1,2,2,0,1,0,1,0,4,4,1,0,4,4,0,0,1,6,1,0,4,6,
%T A382773 4,0,1,6,4,0,1,6,1,0,0,8,1,0,4,0,6,0,1,0,6,0,6,8,1,0,1,10,0,0,8,6,1,0,
%U A382773 8,6,1,0,1,10,0,0,6,6,1,0,0,12,1,0,16
%N A382773 Number of ways to permute a multiset whose multiplicities are the prime indices of n so that the run-lengths are all different.
%C A382773 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 A382773 a(n) = A382771(A181821(n)) = A382771(A304660(n)).
%e A382773 The a(n) partitions for n = 6, 21, 30, 46:
%e A382773   (1,1,2)  (1,1,1,1,2,2)  (1,1,1,2,2,3)  (1,1,1,1,1,1,1,1,1,2)
%e A382773   (2,1,1)  (1,1,1,2,2,1)  (1,1,1,3,2,2)  (1,1,1,1,1,1,1,2,1,1)
%e A382773            (1,2,2,1,1,1)  (2,2,1,1,1,3)  (1,1,1,1,1,1,2,1,1,1)
%e A382773            (2,2,1,1,1,1)  (2,2,3,1,1,1)  (1,1,1,1,1,2,1,1,1,1)
%e A382773                           (3,1,1,1,2,2)  (1,1,1,1,2,1,1,1,1,1)
%e A382773                           (3,2,2,1,1,1)  (1,1,1,2,1,1,1,1,1,1)
%e A382773                                          (1,1,2,1,1,1,1,1,1,1)
%e A382773                                          (2,1,1,1,1,1,1,1,1,1)
%t A382773 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A382773 Table[Length[Select[Permutations[nrmptn[n]],UnsameQ@@Length/@Split[#]&]],{n,100}]
%Y A382773 Positions of 1 are A008578.
%Y A382773 For anti-run permutations we have A335125.
%Y A382773 For just prime indices we have A382771, firsts A382772, equal A382857.
%Y A382773 These permutations for factorials are counted by A382774, equal A335407.
%Y A382773 For equal instead of distinct run-lengths we have A382858.
%Y A382773 Positions of 0 are A382912, complement A382913.
%Y A382773 A044813 lists numbers whose binary expansion has distinct run-lengths, equal A140690.
%Y A382773 A055396 gives least prime index, greatest A061395.
%Y A382773 A056239 adds up prime indices, row sums of A112798.
%Y A382773 A098859 counts partitions with distinct multiplicities, ordered A242882.
%Y A382773 A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
%Y A382773 A329738 counts compositions with equal run-lengths, ranks A353744.
%Y A382773 A329739 counts compositions with distinct run-lengths, ranks A351596.
%Y A382773 A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
%Y A382773 Cf. A000670, A000720, A003242, A048767, A130091, A181821, A238130, A305936, A351013, A351202, A382876, A382879.
%K A382773 nonn
%O A382773 1,6
%A A382773 _Gus Wiseman_, Apr 09 2025