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

%I A382771 #16 Apr 21 2025 10:47:19
%S A382771 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,
%T A382771 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,
%U A382771 0,0,1,2,1,0,2,2,0,0,1,2,1,0,1,0,0,0,0
%N A382771 Number of ways to permute the prime indices of n so that the run-lengths are all different.
%C A382771 The first x with a(x) > 0 but A382857(x) > 1 is a(216) = 4, A382857(216) = 4.
%C A382771 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.
%F A382771 a(A181821(n)) = a(A304660(n)) = A382773(n).
%F A382771 a(n!) = A382774(n).
%e A382771 The a(96) = 4 permutations are:
%e A382771   (1,1,1,1,1,2)
%e A382771   (1,1,1,2,1,1)
%e A382771   (1,1,2,1,1,1)
%e A382771   (2,1,1,1,1,1)
%e A382771 The a(216) = 4 permutations are:
%e A382771   (1,1,2,2,2,1)
%e A382771   (1,2,2,2,1,1)
%e A382771   (2,1,1,1,2,2)
%e A382771   (2,2,1,1,1,2)
%e A382771 The a(360) = 6 permutations are:
%e A382771   (1,1,1,2,2,3)
%e A382771   (1,1,1,3,2,2)
%e A382771   (2,2,1,1,1,3)
%e A382771   (2,2,3,1,1,1)
%e A382771   (3,1,1,1,2,2)
%e A382771   (3,2,2,1,1,1)
%t A382771 Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]],UnsameQ@@Length/@Split[#]&]],{n,30}]
%Y A382771 Positions of 1 are A000961.
%Y A382771 Positions of positive terms are A351294, conjugate A381432.
%Y A382771 Positions of 0 are A351295, conjugate A381433, equal A382879.
%Y A382771 Sorted positions of first appearances are A382772, equal A382878.
%Y A382771 For prescribed signature we have A382773, equal A382858.
%Y A382771 The restriction to factorials is A382774, equal A335407.
%Y A382771 For equal instead of distinct run-lengths we have A382857.
%Y A382771 For run-sums instead of run-lengths we have A382876, equal A382877.
%Y A382771 Positions of terms > 1 are A383113.
%Y A382771 A044813 lists numbers whose binary expansion has distinct run-lengths.
%Y A382771 A055396 gives least prime index, greatest A061395.
%Y A382771 A056239 adds up prime indices, row sums of A112798.
%Y A382771 A098859 counts partitions with distinct multiplicities, ordered A242882.
%Y A382771 A239455 counts Look-and-Say partitions, complement A351293.
%Y A382771 A329738 counts compositions with equal run-lengths, ranks A353744.
%Y A382771 A329739 counts compositions with distinct run-lengths, ranks A351596.
%Y A382771 Cf. A000720, A001221, A001222, A003242, A048767, A051903, A051904, A130091, A238130, A351013, A351202.
%K A382771 nonn
%O A382771 1,12
%A A382771 _Gus Wiseman_, Apr 07 2025