A386638 Number of integer partitions of n of inseparable type.

0, 0, 1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 19, 19, 30, 30, 45, 45, 67, 67, 97, 97, 139, 139, 195, 195, 272, 272, 373, 373, 508, 508, 684, 684, 915, 915, 1212, 1212, 1597, 1597, 2087, 2087, 2714, 2714, 3506, 3506, 4508, 4508, 5763, 5763, 7338, 7338, 9296, 9296

Offset: 0

Author

Gus Wiseman, Aug 14 2025

Keywords

• nonn
• new

Comments

A multiset is inseparable iff it has no permutation without adjacent equal parts. It is of inseparable type iff any multiset with those multiplicities (type) is inseparable. For example, {1,1,2} is separable but {1,1,1,2} is not; hence (2,1) is of separable type but (3,1) is not.

Also the number of integer partitions of n whose greatest part is at least two more than the sum of all the other parts.

Examples

			The a(2) = 1 through a(10) = 12 partitions (A=10):
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)     (A)
            (31)  (41)  (42)   (52)   (53)    (63)    (64)
                        (51)   (61)   (62)    (72)    (73)
                        (411)  (511)  (71)    (81)    (82)
                                      (521)   (621)   (91)
                                      (611)   (711)   (622)
                                      (5111)  (6111)  (631)
                                                      (721)
                                                      (811)
                                                      (6211)
                                                      (7111)
                                                      (61111)
		

Crossrefs

Reduplication of A000070 shifted right.

Same as A025065 shifted right twice.

The Heinz numbers of these partitions are A335126.

Row sums of A386586.

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

A239455 counts Look-and-Say partitions, inseparable case A386632.

A325534 counts separable multisets, ranks A335433, sums of A386583.

A325535 counts inseparable multisets, ranks A335448, sums of A386584.

A335434 counts separable factorizations, inseparable A333487.

A336103 counts normal separable multisets, inseparable A336102.

A336106 counts separable type partitions, ranks A335127, sums of A386585.

A386633 counts separable type set partitions, row sums of A386635.

A386634 counts inseparable type set partitions, row sums of A386636.

Cf. A000110, A008284, A047966, A072233, A106351, A124762, A217605, A239964, A279790, A335125, A351293, A386577.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],2*Max@@#>1+n&]],{n,0,15}]

Formula

For n>1, a(n) = A025065(n-2).

a(n) = A000041(n) - A336106(n).