A386586 Triangle read by rows where T(n,k) is the number of integer partitions y of n into k parts such that any multiset whose multiplicities are the parts of y is inseparable.

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 3, 2, 1, 0, 0, 0, 0, 0, 1, 3, 2, 1, 0, 0, 0, 0, 0, 0, 1, 4, 4, 2, 1, 0, 0, 0, 0, 0

Offset: 0

Author

Gus Wiseman, Aug 05 2025

Keywords

• nonn
• tabl
• more

Comments

We say that such partitions are of inseparable type. This is different from inseparable partitions (see A386584). A multiset is separable iff it has a permutation without any adjacent equal parts.

Examples

			The partition y = (7,2,1) is the multiplicities of the multiset {1,1,1,1,1,1,1,2,2,3}, which is inseparable, so y is counted under T(10,3).
Row n = 10 counts the following partitions (A = 10):
  .  A  91  811  7111  61111  .  .  .  .  .
        82  721  6211
        73  631
        64  622
Triangle begins:
  0
  0 0
  0 1 0
  0 1 0 0
  0 1 1 0 0
  0 1 1 0 0 0
  0 1 2 1 0 0 0
  0 1 2 1 0 0 0 0
  0 1 3 2 1 0 0 0 0
  0 1 3 2 1 0 0 0 0 0
  0 1 4 4 2 1 0 0 0 0 0
		

Crossrefs

This is the inseparable type case of A008284 or A072233.

Row sums shifted left once are A025065 (ranks A335126), separable version A336106 (ranks A335127).

For separable instead of inseparable type we have A386583.

For integer partitions instead of normal multisets we have A386584.

For separable type instead of inseparable type we have A386585.

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

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

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.

Cf. A000670, A005651, A106351, A238130, A279790, A386575, A386576, A386579, A386582, A386634.

Programs

Mathematica

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

Formula

a(n) = A072233(n) - A386585(n).