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

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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Jul 31 2025

Keywords

• nonn

Comments

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.

Examples

			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.
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.
		

Crossrefs

For separable instead of inseparable we have A386575.

This is the inseparable case of A386587 (ordered version A382525).

Positions of positive terms are A386632.

Positions of first appearances are A386637.

A000110 counts set partitions, ordered A000670.

A003242 and A335452 count separations, ranks A333489.

A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.

A239455 counts Look-and-Say partitions (ranks A351294), complement A351293 (ranks A351295).

A279790 counts disjoint families on strongly normal multisets.

A325534 counts separable multisets, ranks A335433, sums of A386583.

A325535 counts inseparable multisets, ranks A335448, sums of A386584.

A336106 counts partitions of separable type, ranks A335127, sums of A386585.

A386633 counts separable set partitions, row sums of A386635.

A386634 counts inseparable set partitions, row sums of A386636.

Cf. A001221, A001222, A008480, A051903, A051904, A056239, A111133, A130091, A373957, A386580, A386581.

Programs

Mathematica

disjointFamilies[y_]:=Union[Sort/@Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&]];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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&]}]];
seps[ptn_]:=If[Total[ptn]==0,{{}},Join@@(seps[ptn,#]&/@Range[Length[ptn]])];
Table[Length[Select[disjointFamilies[prix[n]],seps[Length/@#]=={}&]],{n,100}]

Formula

a(2^n) = A111133(n).