A386576 -id:A386576 - cp's OEIS frontend

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

Gus Wiseman, Aug 03 2025

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

			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
  ...

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

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.

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

A386584 Triangle read by rows where T(n,k) is the number of length k>=0 integer partitions of n having no permutation without any adjacent equal parts (inseparable).

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Aug 05 2025

A multiset is inseparable iff it has no anti-run permutations, where an anti-run is a sequence without any adjacent equal parts. Inseparable partitions (A325535) are different from partitions of inseparable type (A386586).

			Row n = 10 counts the following partitions:
  . . 55 . 7111 61111 511111 4111111 31111111 211111111 1111111111
           4222 22222 421111 3211111 22111111
           3331       331111
                      222211
Triangle begins:
  0
  0  0
  0  0  1
  0  0  0  1
  0  0  1  0  1
  0  0  0  0  1  1
  0  0  1  1  1  1  1
  0  0  0  0  2  1  1  1
  0  0  1  0  2  1  2  1  1
  0  0  0  1  2  2  2  2  1  1
  0  0  1  0  3  2  4  2  2  1  1
  0  0  0  0  3  2  4  3  3  2  1  1
  0  0  1  1  3  2  6  4  4  3  2  1  1
  0  0  0  0  4  3  6  5  6  4  3  2  1  1
  0  0  1  0  4  3  9  6  8  5  5  3  2  1  1
  0  0  0  1  4  3  9  7 10  8  6  5  3  2  1  1
  0  0  1  0  5  3 12  8 13  9 10  6  5  3  2  1  1
  0  0  0  0  5  4 12 10 16 12 12  9  7  5  3  2  1  1
  0  0  1  1  5  4 16 11 20 15 17 12 10  7  5  3  2  1  1
  0  0  0  0  6  4 16 13 24 18 21 16 14 10  7  5  3  2  1  1
  0  0  1  0  6  4 20 14 29 21 28 20 19 13 11  7  5  3  2  1  1

Inseparable case of A008284 or A072233.
Row sums are A325535, ranked by A335448.
For separable instead of inseparable we have A386583, sums A325534, ranks A335433.
For separable type we have A386585, sums A336106, ranks A335127.
For inseparable type 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.
A336103 counts normal separable multisets, inseparable A336102.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A106351, A111133, A238130, A239455, A335434, A351293, A386575, A386576, A386580, A386581, A386577.

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

T(n,k) = A072233(n,k) - A386583(n,k).

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.

Original entry on oeis.org

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

Gus Wiseman, Aug 02 2025

We say that such partitions are of separable type.

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

			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

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.

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}]

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

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.

Original entry on oeis.org

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

Gus Wiseman, Aug 05 2025

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.

			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

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.

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}]

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

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A386584 Triangle read by rows where T(n,k) is the number of length k>=0 integer partitions of n having no permutation without any adjacent equal parts (inseparable).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula