A325243 Number of integer partitions of n with exactly two distinct multiplicities.

0, 0, 0, 0, 1, 3, 3, 9, 12, 19, 26, 39, 47, 70, 89, 115, 148, 189, 235, 294, 362, 450, 558, 669, 817, 980, 1197, 1421, 1709, 2012, 2429, 2836, 3380, 3961, 4699, 5433, 6457, 7433, 8770, 10109, 11818, 13547, 15912, 18109, 21105, 24121, 27959, 31736, 36840, 41670

Offset: 0

Author

Gus Wiseman, Apr 15 2019

Keywords

• nonn

Comments

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

The Heinz numbers of these partitions are given by A323055.

Examples

			The a(4) = 1 through a(9) = 19 partitions:
  (211)  (221)   (411)    (322)     (332)      (441)
         (311)   (3111)   (331)     (422)      (522)
         (2111)  (21111)  (511)     (611)      (711)
                          (2221)    (3221)     (3222)
                          (3211)    (4211)     (3321)
                          (4111)    (5111)     (4221)
                          (22111)   (22211)    (4311)
                          (31111)   (32111)    (5211)
                          (211111)  (41111)    (6111)
                                    (221111)   (22221)
                                    (311111)   (32211)
                                    (2111111)  (33111)
                                               (42111)
                                               (51111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
		

Crossrefs

Column k = 2 of A325242. Dominated by A325267.

Cf. A008284, A062770, A071625, A098859, A116608, A244515, A323055, A325244.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Length[Union[Length/@Split[#]]]==2&]],{n,0,30}]