cp's OEIS Frontend

A386582 Number of distinct inseparable and pairwise disjoint sets of strict integer partitions, one of each exponent in the prime factorization of n.

%I A386582 #7 Aug 02 2025 21:54:45
%S A386582 0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,2,0,0,
%T A386582 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,
%U A386582 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0
%N A386582 Number of distinct inseparable and pairwise disjoint sets of strict integer partitions, one of each exponent in the prime factorization of n.
%C A386582 A set partition is inseparable iff the underlying set has no permutation whose adjacent elements all belong to different blocks. Note that inseparability only depends on the sizes of the blocks.
%F A386582 a(2^n) = A111133(n).
%e A386582 The prime indices of 9216 are {1,1,1,1,1,1,1,1,1,1,2,2}, with a(9216) = 2 choices: {{2},{1,4,5}} and {{2},{1,3,6}}. The other 4 disjoint families {{2},{10}}, {{2},{4,6}}, {{2},{3,7}}, {{2},{1,9}} are separable.
%e A386582 The prime indices of 15552 are {1,1,1,1,1,1,2,2,2,2,2}, with a(15552) = 1 choice: {{5},{1,2,3}}. The other 5 disjoint families {{5},{6}}, {{5},{2,4}}, {{6},{2,3}}, {{6},{1,4}}, {{1,5},{2,3}} are separable.
%t A386582 disjointFamilies[y_]:=Union[Sort/@Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&]];
%t A386582 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A386582 seps[ptn_,fir_]:=If[Total[ptn]==1,{{fir}},Join@@Table[Prepend[#,fir]&/@seps[MapAt[#-1&,ptn,fir],nex],{nex,Select[DeleteCases[Range[Length[ptn]],fir],ptn[[#]]>0&]}]];
%t A386582 seps[ptn_]:=If[Total[ptn]==0,{{}},Join@@(seps[ptn,#]&/@Range[Length[ptn]])];
%t A386582 Table[Length[Select[disjointFamilies[prix[n]],seps[Length/@#]=={}&]],{n,100}]
%Y A386582 For separable instead of inseparable we have A386575.
%Y A386582 This is the inseparable case of A386587 (ordered version A382525).
%Y A386582 Positions of positive terms are A386632.
%Y A386582 Positions of first appearances are A386637.
%Y A386582 A000110 counts set partitions, ordered A000670.
%Y A386582 A003242 and A335452 count separations, ranks A333489.
%Y A386582 A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
%Y A386582 A239455 counts Look-and-Say partitions (ranks A351294), complement A351293 (ranks A351295).
%Y A386582 A279790 counts disjoint families on strongly normal multisets.
%Y A386582 A325534 counts separable multisets, ranks A335433, sums of A386583.
%Y A386582 A325535 counts inseparable multisets, ranks A335448, sums of A386584.
%Y A386582 A336106 counts partitions of separable type, ranks A335127, sums of A386585.
%Y A386582 A386633 counts separable set partitions, row sums of A386635.
%Y A386582 A386634 counts inseparable set partitions, row sums of A386636.
%Y A386582 Cf. A001221, A001222, A008480, A051903, A051904, A056239, A111133, A130091, A373957, A386580, A386581.
%K A386582 nonn
%O A386582 1,32
%A A386582 _Gus Wiseman_, Jul 31 2025