cp's OEIS Frontend

A386575 Number of distinct separable and pairwise disjoint sets of strict integer partitions, one of each exponent in the prime factorization of n.

%I A386575 #14 Aug 02 2025 21:54:53
%S A386575 1,1,1,1,1,0,1,1,1,0,1,1,1,0,0,1,1,1,1,1,0,0,1,1,1,0,1,1,1,0,1,1,0,0,
%T A386575 0,0,1,0,0,1,1,0,1,1,1,0,1,1,1,1,0,1,1,1,0,1,0,0,1,0,1,0,1,1,0,0,1,1,
%U A386575 0,0,1,1,1,0,1,1,0,0,1,1,1,0,1,0,0,0,0,1,1,0,0,1,0,0,0,2
%N A386575 Number of distinct separable and pairwise disjoint sets of strict integer partitions, one of each exponent in the prime factorization of n.
%C A386575 A set partition is separable iff the underlying set has a permutation whose adjacent elements all belong to different blocks. Note that separability only depends on the sizes of the blocks.
%C A386575 Conjecture: a(n) > 0 iff the multiset of prime factors of n has a permutation with all distinct run lengths.
%e A386575 The prime indices of 6144 are {1,1,1,1,1,1,1,1,1,1,1,2}, and we have the following a(6144) = 5 choices: {{1},{11}}, {{1},{5,6}}, {{1},{4,7}}, {{1},{3,8}}, {{1},{2,9}}. The other 2 disjoint families {{1},{2,4,5}} and {{1},{2,3,6}} are not separable.
%e A386575 The prime indices of 7776 are {1,1,1,1,1,2,2,2,2,2}, with separable disjoint families {{5},{2,3}}, {{5},{1,4}}, {{1,4},{2,3}}, so a(7776) = 3.
%e A386575 The prime indices of 15552 are {1,1,1,1,1,1,2,2,2,2,2}, with a(15552) = 5 choices: {{5},{6}}, {{5},{2,4}}, {{6},{2,3}}, {{6},{1,4}}, {{1,5},{2,3}}. The other disjoint family {{5},{1,2,3}} is not separable.
%e A386575 The a(n) families for n = 2, 96, 384, 1536, 3456, 20736:
%e A386575   {{1}}  {{1},{5}}    {{1},{7}}    {{1},{9}}    {{3},{7}}      {{4},{8}}
%e A386575          {{1},{2,3}}  {{1},{2,5}}  {{1},{2,7}}  {{3},{1,6}}    {{4},{1,7}}
%e A386575                       {{1},{3,4}}  {{1},{3,6}}  {{3},{2,5}}    {{4},{2,6}}
%e A386575                                    {{1},{4,5}}  {{7},{1,2}}    {{4},{3,5}}
%e A386575                                                 {{1,2},{3,4}}  {{8},{1,3}}
%e A386575                                                                {{1,3},{2,6}}
%t A386575 disjointFamilies[y_]:=Union[Sort/@Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&]];
%t A386575 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A386575 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 A386575 seps[ptn_]:=If[Total[ptn]==0,{{}},Join@@(seps[ptn,#]&/@Range[Length[ptn]])];
%t A386575 Table[Length[Select[disjointFamilies[prix[n]],seps[Length/@#]!={}&]],{n,100}]
%Y A386575 Positions of positive terms are A351294, conjugate A381432.
%Y A386575 Positions of 0 are A351295, conjugate A381433.
%Y A386575 For inseparable instead of separable we have A386582, see A386632.
%Y A386575 This is the separable case of A386587 (ordered version A382525).
%Y A386575 A000110 counts set partitions, ordered A000670.
%Y A386575 A003242 and A335452 count separations, ranks A333489.
%Y A386575 A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
%Y A386575 A239455 counts Look-and-Say partitions, complement A351293.
%Y A386575 A279790 counts disjoint families on strongly normal multisets.
%Y A386575 A325534 counts separable multisets, ranks A335433, sums of A386583.
%Y A386575 A325535 counts inseparable multisets, ranks A335448, sums of A386584.
%Y A386575 A336106 counts partitions of separable type, ranks A335127, sums of A386585.
%Y A386575 A386633 counts separable set partitions, row sums of A386635.
%Y A386575 A386634 counts inseparable set partitions, row sums of A386636.
%Y A386575 Cf. A001221, A001222, A008480, A051903, A051904, A056239, A130091, A373957, A386580, A386581.
%K A386575 nonn
%O A386575 1,96
%A A386575 _Gus Wiseman_, Jul 30 2025