A336106 Number of integer partitions of n whose greatest part is at most one more than the sum of the other parts.

1, 1, 1, 2, 3, 5, 7, 11, 15, 23, 30, 44, 58, 82, 105, 146, 186, 252, 318, 423, 530, 695, 863, 1116, 1380, 1763, 2164, 2738, 3345, 4192, 5096, 6334, 7665, 9459, 11395, 13968, 16765, 20425, 24418, 29588, 35251, 42496, 50460, 60547, 71669, 85628

Offset: 0

Author

Gus Wiseman, Jul 09 2020

Keywords

• nonn

Comments

Also the number of separable strong multisets of length n covering an initial interval of positive integers. A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

Examples

			The a(1) = 1 through a(8) = 15 partitions:
  (1)  (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (211)   (221)    (222)     (322)      (332)
                    (1111)  (311)    (321)     (331)      (422)
                            (2111)   (2211)    (421)      (431)
                            (11111)  (3111)    (2221)     (2222)
                                     (21111)   (3211)     (3221)
                                     (111111)  (4111)     (3311)
                                               (22111)    (4211)
                                               (31111)    (22211)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
		

Crossrefs

The inseparable version is A025065.

The Heinz numbers of these partitions are A335127.

The non-strong version is A336103.

Sequences covering an initial interval are A000670.

Anti-run compositions are A003242.

Anti-run patterns are A005649.

Separable partitions are A325534.

Inseparable partitions are A325535.

Separable factorizations are A335434.

Heinz numbers of separable partitions are A335433.

Cf. A049610, A106351, A292884, A335126, A335433, A335452, A335548, A336102.

Programs

Mathematica

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