A386583 Triangle read by rows where T(n,k) is the number of length k integer partitions of n having a permutation without any adjacent equal parts (separable).

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 2, 2, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 1, 3, 4, 1, 1, 0, 0, 0, 1, 3, 5, 3, 2, 0, 0, 0, 0, 1, 4, 6, 4, 3, 1, 0, 0, 0, 0, 1, 4, 8, 6, 5, 1, 1, 0, 0, 0, 0, 1, 5, 10, 8, 8, 3, 2, 0, 0, 0, 0, 0, 1, 5, 11, 12, 11, 5, 3, 1, 0, 0, 0, 0

Offset: 0

Author

Gus Wiseman, Aug 03 2025

Keywords

• nonn
• tabl

Comments

A multiset is separable iff it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

Separable partitions (A325534) are different from partitions of separable type (A386585).

Are the rows all unimodal?

Some rows are not unimodal: T(200, k=26..30) = 149371873744, 153304102463, 152360653274, 152412869411, 147228477998. - Alois P. Heinz, Aug 04 2025

Examples

			Row n = 9 counts the following partitions:
  (9)  (5,4)  (4,3,2)  (3,3,2,1)  (3,2,2,1,1)  (2,2,2,1,1,1)
       (6,3)  (4,4,1)  (4,2,2,1)  (3,3,1,1,1)
       (7,2)  (5,2,2)  (4,3,1,1)  (4,2,1,1,1)
       (8,1)  (5,3,1)  (5,2,1,1)
              (6,2,1)
              (7,1,1)
Triangle begins:
  1
  0  1
  0  1  0
  0  1  1  0
  0  1  1  1  0
  0  1  2  2  0  0
  0  1  2  2  1  0  0
  0  1  3  4  1  1  0  0
  0  1  3  5  3  2  0  0  0
  0  1  4  6  4  3  1  0  0  0
  0  1  4  8  6  5  1  1  0  0  0
  0  1  5 10  8  8  3  2  0  0  0  0
  0  1  5 11 12 11  5  3  1  0  0  0  0
  0  1  6 14 14 15  8  6  1  1  0  0  0  0
  0  1  6 16 19 20 11  9  3  2  0  0  0  0  0
  0  1  7 18 23 27 17 14  5  3  1  0  0  0  0  0
  0  1  7 21 29 34 23 20  9  6  1  1  0  0  0  0  0
  0  1  8 24 34 43 32 28 13 10  3  2  0  0  0  0  0  0
  0  1  8 26 42 53 42 38 20 15  5  3  1  0  0  0  0  0  0
  0  1  9 30 48 66 55 52 28 23  9  6  1  1  0  0  0  0  0  0
  0  1  9 33 58 80 70 68 41 33 14 10  3  2  0  0  0  0  0  0  0
  ...
		

Links

• Alois P. Heinz, Rows n = 0..200, flattened

Crossrefs

Separable case of A008284.

Row sums are A325534, ranked by A335433.

For inseparable instead separable we have A386584, sums A325535, ranks A335448.

For separable type instead of separable we have A386585, sums A336106, ranks A335127.

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

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

A124762 gives inseparability of standard compositions, separability A333382.

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

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. A106351, A111133, A238130, A335434, A386575, A386576, A386580, A386581, A386577.

Programs

Mathematica

sepQ[y_]:=Select[Permutations[y],Length[Split[#]]==Length[y]&]!={};
Table[Length[Select[IntegerPartitions[n,{k}],sepQ]],{n,0,15},{k,0,n}]