A325244 Number of integer partitions of n with one fewer distinct multiplicities than distinct parts.

0, 0, 0, 1, 1, 2, 3, 4, 7, 12, 16, 21, 33, 38, 50, 75, 87, 111, 150, 185, 244, 307, 373, 461, 585, 702, 856, 1043, 1255, 1498, 1822, 2143, 2565, 3064, 3607, 4251, 5064, 5920, 6953, 8174, 9503, 11064, 12927, 14921, 17320, 19986, 23067, 26485, 30499, 34894

Offset: 0

Author

Gus Wiseman, Apr 15 2019

Keywords

• nonn

Comments

For example, (32211) has two distinct multiplicities (1, 2) and three distinct parts (1, 2, 3) so is counted under a(9).

The Heinz numbers of these partitions are given by A325259.

Examples

			The a(3) = 1 through a(10) = 16 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)     (54)      (64)
              (41)  (51)    (52)    (62)     (63)      (73)
                    (2211)  (61)    (71)     (72)      (82)
                            (3211)  (3221)   (81)      (91)
                                    (3311)   (3321)    (3322)
                                    (4211)   (4221)    (4411)
                                    (32111)  (4311)    (5221)
                                             (5211)    (5311)
                                             (32211)   (6211)
                                             (42111)   (32221)
                                             (222111)  (33211)
                                             (321111)  (42211)
                                                       (43111)
                                                       (52111)
                                                       (421111)
                                                       (3211111)
		

Crossrefs

Cf. A008284, A071625, A090858, A098859, A116608, A117571, A127002, A325242, A325259, A325270.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==Length[Union[Length/@Split[#]]]+1&]],{n,0,30}]