A325279 Number of integer partitions of n whose maximum multiplicity is one greater than their minimum multiplicity.

0, 0, 0, 0, 1, 2, 1, 5, 6, 9, 10, 18, 18, 31, 34, 48, 57, 80, 86, 122, 138, 183, 211, 275, 311, 402, 461, 576, 663, 825, 942, 1163, 1334, 1621, 1865, 2248, 2566, 3084, 3532, 4193, 4794, 5674, 6472, 7617, 8685, 10153, 11576, 13483, 15320, 17790, 20200, 23342

Offset: 0

Author

Gus Wiseman, Apr 18 2019

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A325241.

For example, the partition (44111) has two multiplicities (2 and 3) which differ by 1, so is counted under a(11).

Examples

			The a(4) = 1 through a(11) = 18 partitions:
  (211)  (221)  (411)  (322)    (332)    (441)    (433)      (443)
         (311)         (331)    (422)    (522)    (442)      (533)
                       (511)    (611)    (711)    (622)      (551)
                       (3211)   (3221)   (3321)   (811)      (722)
                       (22111)  (4211)   (4221)   (5221)     (911)
                                (22211)  (4311)   (5311)     (4322)
                                         (5211)   (6211)     (4331)
                                         (32211)  (33211)    (4421)
                                         (33111)  (42211)    (5411)
                                                  (2221111)  (6221)
                                                             (6311)
                                                             (7211)
                                                             (33221)
                                                             (33311)
                                                             (43211)
                                                             (44111)
                                                             (52211)
                                                             (2222111)
		

Crossrefs

Cf. A047966, A062770, A071625, A098859, A117571, A127002, A325242, A325244, A325245, A325280.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Max@@Length/@Split[#]-Min@@Length/@Split[#]==1&]],{n,0,30}]