A386585 Triangle read by rows where T(n,k) is the number of integer partitions y of n into k = 0..n parts such that any multiset whose multiplicities are the parts of y is separable.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 1, 1, 0, 0, 1, 2, 2, 1, 1, 0, 0, 1, 3, 3, 2, 1, 1, 0, 0, 1, 3, 4, 3, 2, 1, 1, 0, 0, 1, 5, 5, 5, 3, 2, 1, 1, 0, 0, 1, 4, 7, 6, 5, 3, 2, 1, 1

Offset: 0

Author

Gus Wiseman, Aug 02 2025

Keywords

• nonn
• tabl
• more

Comments

We say that such partitions are of separable type.

A multiset is separable iff it has a permutation without any adjacent equal parts.

Examples

			Row n = 8 counts the following partitions:
  .  .  44  431  4211  41111  311111  2111111  11111111
            422  3311  32111  221111
            332  3221  22211
                 2222
with the following separable multisets:
  . . 11112222 11112223 11112234 11112345 11123456 11234567 12345678
               11112233 11122234 11122345 11223456
               11122233 11122334 11223345
                        11223344
Triangle begins:
  1
  0  1
  0  0  1
  0  0  1  1
  0  0  1  1  1
  0  0  1  2  1  1
  0  0  1  2  2  1  1
  0  0  1  3  3  2  1  1
  0  0  1  3  4  3  2  1  1
  0  0  1  5  5  5  3  2  1  1
  0  0  1  4  7  6  5  3  2  1  1
		

Crossrefs

This is the separable type case of A072233 or A008284.

Row sums are A336106, ranks A335127.

For separable instead of separable type we have A386583, inseparable A386584.

For inseparable instead of separable we have A386586, sums A025065, ranks A335126.

A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.

A239455 counts Look-and-Say partitions, ranks A351294.

A279790 counts disjoint families on strongly normal multisets.

A325534 counts separable multisets, ranks A335433.

A325535 counts inseparable multisets, ranks A335448.

A336103 counts normal separable multisets, inseparable A336102.

A351293 counts non-Look-and-Say partitions, ranks A351295.

A386633 counts separable set partitions, row sums of A386635.

A386634 counts inseparable set partitions, row sums of A386636.

Cf. A005651, A106351, A111133, A238130, A335434, A386575, A386576, A386579, A386580, A386581, A386582.

Programs

Mathematica

sepQ[y_]:=Select[Permutations[y],Length[Split[#]]==Length[y]&]!={};
mst[y_]:=Join@@Table[ConstantArray[k,y[[k]]],{k,Length[y]}];
Table[Length[Select[IntegerPartitions[n,{k}],sepQ[mst[#]]&]],{n,0,5},{k,0,n}]

Formula

a(n) = A072233(n) - A386586(n).