Find disjoint subsets and their union is {1...9}

Question

I have such a list

sets = Select[IntegerDigits /@ (Range[Floor@Sqrt@987654321]^2), 
  DuplicateFreeQ@# && FreeQ[#, 0] &]

I want to pick all subsets of them that non intersect and their union is ${1...9}$. For example:

samples={{{9},{1,3,5,2,7,6,8,4}},{{1},{4},{3,2,9,7,8,5,6}},
  {{1},{4},{9},{8,7,2,3,5,6}},{{1},{9},{2,5},{3,6},{7,8,4}}};

I came up with

n=3;
Select[Subsets[sets, {n}], Sort[Join@@#]==Range[9]&]//AbsoluteTiming

For n>3, it's too slow, Is there a more efficient way?

Answer 1

sets = Select[IntegerDigits /@ (Range[Floor@Sqrt@987654321]^2), 
   DuplicateFreeQ@# && FreeQ[#, 0] &];

list = Union[Sort /@ 
  Flatten[Select[Tuples[(d |-> Select[sets, Length[#] == d &]) /@ #], 
      Length[Union[Flatten[#]]] == 9 &] & /@ IntegerPartitions[9], 1]]

Length[list]

---

{{{1, 3, 9, 8, 5, 4, 2, 7, 6}}, {{1, 5, 2, 8, 4, 3, 7, 6, 9}},(*...*),
{{9}, {8, 1}, {3, 2, 4}, {5, 7, 6}}, {{1}, {9}, {2, 5}, {3, 6}, {7, 8, 4}}}

82

Lists in number form:

Map[FromDigits, list, {2}]

---

{{139854276}, {152843769}, {157326849}, {215384976}, {245893761}, \
{254817369}, {326597184}, {361874529}, {375468129}, {382945761}, \
{385297641}, {412739856}, {523814769}, {529874361}, {537219684}, \
{549386721}, {587432169}, {589324176}, {597362481}, {615387249}, \
{627953481}, {653927184}, {672935481}, {697435281}, {714653289}, \
{735982641}, {743816529}, {842973156}, {847159236}, {923187456}, {9, 
  13527684}, {9, 34857216}, {9, 65318724}, {9, 73256481}, {9, 
  81432576}, {36, 5184729}, {36, 5948721}, {81, 2537649}, {81, 
  5673924}, {81, 7436529}, {81, 9253764}, {324, 751689}, {576, 
  321489}, {576, 349281}, {576, 381924}, {729, 385641}, {3249, 
  15876}, {4356, 71289}, {8649, 35721}, {1, 4, 3297856}, {1, 4, 
  3857296}, {1, 4, 5827396}, {1, 4, 6385729}, {1, 4, 8567329}, {1, 4, 
  9572836}, {1, 49, 872356}, {1, 64, 537289}, {1, 256, 73984}, {1, 
  625, 73984}, {4, 16, 537289}, {4, 25, 139876}, {4, 25, 391876}, {4, 
  289, 15376}, {9, 324, 15876}, {16, 25, 73984}, {16, 784, 5329}, {25,
   784, 1369}, {25, 784, 1936}, {25, 841, 7396}, {36, 81, 74529}, {36,
   81, 79524}, {36, 729, 5184}, {81, 324, 7569}, {81, 576, 3249}, {81,
   729, 4356}, {361, 529, 784}, {1, 4, 9, 872356}, {1, 36, 529, 
  784}, {4, 25, 81, 7396}, {9, 25, 361, 784}, {9, 81, 324, 576}, {1, 
  9, 25, 36, 784}}