cp's OEIS Frontend

A382876 Number of ways to permute the prime indices of n so that the run-sums are all different.

%I A382876 #22 Apr 27 2025 09:09:03
%S A382876 1,1,1,1,1,2,1,1,1,2,1,0,1,2,2,1,1,2,1,2,2,2,1,2,1,2,1,2,1,6,1,1,2,2,
%T A382876 2,2,1,2,2,2,1,6,1,2,2,2,1,4,1,2,2,2,1,4,2,4,2,2,1,0,1,2,0,1,2,6,1,2,
%U A382876 2,6,1,4,1,2,2,2,2,6,1,2,1,2,1,0,2,2,2
%N A382876 Number of ways to permute the prime indices of n so that the run-sums are all different.
%C A382876 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.
%C A382876 A run in a sequence is a constant consecutive subsequence. The run-sums of a sequence are obtained by splitting it into maximal runs and taking their sums. See A353932 for run-sums of standard compositions.
%e A382876 For n = 12, none of the permutations (1,1,2), (1,2,1), (2,1,1) has distinct run-sums, so a(12) = 0.
%e A382876 The prime indices of 36 are {1,1,2,2}, and we have permutations: (1,1,2,2), (2,2,1,1), so a(36) = 2.
%e A382876 For n = 90 we have:
%e A382876   (1,2,2,3)
%e A382876   (1,3,2,2)
%e A382876   (2,2,1,3)
%e A382876   (2,2,3,1)
%e A382876   (3,1,2,2)
%e A382876   (3,2,2,1)
%e A382876 So a(90) = 6. The 6 missing permutations are: (1,2,3,2), (2,1,2,3), (2,1,3,2), (2,3,1,2), (2,3,2,1), (3,2,1,2).
%t A382876 Table[Length[Select[Permutations[PrimePi /@ Join@@ConstantArray@@@FactorInteger[n]], UnsameQ@@Total/@Split[#]&]],{n,100}]
%Y A382876 Positions of 1 are A000961.
%Y A382876 Compositions of this type are counted by A353850, ranked by A353852.
%Y A382876 Positions of 0 appear to be A381636, for equal run-sums A383100.
%Y A382876 For run-lengths instead of sums we have A382771, equal A382857 (zeros A382879).
%Y A382876 For equal instead of distinct run-sums we have A382877.
%Y A382876 A044813 lists numbers whose binary expansion has distinct run-lengths.
%Y A382876 A056239 adds up prime indices, row sums of A112798.
%Y A382876 A304442 counts compositions with equal run-sums, complement A382076.
%Y A382876 A329739 counts compositions with distinct run-lengths, ranks A351596.
%Y A382876 A353837 counts partitions with distinct run-sums, ranks A353838.
%Y A382876 A353847 gives composition run-sum transformation, for partitions A353832.
%Y A382876 A353932 lists run-sums of standard compositions.
%Y A382876 Cf. A000720, A001221, A001222, A098859, A130091, A329738, A351013, A351202, A353848, A353851, A354580, A354584.
%K A382876 nonn
%O A382876 1,6
%A A382876 _Gus Wiseman_, Apr 12 2025