A323531 Number of square multiset partitions of integer partitions of n.

1, 1, 1, 1, 2, 2, 4, 5, 9, 12, 18, 24, 36, 48, 69, 97, 139, 196, 283, 402, 576, 819, 1161, 1635, 2301, 3209, 4469, 6193, 8571, 11812, 16291, 22404, 30850, 42414, 58393, 80305, 110578, 152091, 209308, 287686, 395352, 542413, 743603, 1017489, 1390510, 1896482

Offset: 0

Author

Gus Wiseman, Jan 21 2019

Keywords

• nonn

Comments

A multiset partition is square if the number of parts is equal to the number of parts in each part.

Examples

			The a(3) = 1 through a(9) = 12 square multiset partitions:
  (3)  (4)       (5)       (6)       (7)       (8)       (9)
       (11)(11)  (21)(11)  (21)(21)  (22)(21)  (22)(22)  (32)(22)
                           (22)(11)  (31)(21)  (31)(22)  (32)(31)
                           (31)(11)  (32)(11)  (31)(31)  (33)(21)
                                     (41)(11)  (32)(21)  (41)(22)
                                               (33)(11)  (41)(31)
                                               (41)(21)  (42)(21)
                                               (42)(11)  (43)(11)
                                               (51)(11)  (51)(21)
                                                         (52)(11)
                                                         (61)(11)
                                                         (111)(111)(111)
		

Crossrefs

Cf. A000219, A001970, A047968, A261049, A279787, A305551, A319066, A323580.

Programs

Mathematica

Table[Sum[Length[Union@@(Union[Sort/@Tuples[IntegerPartitions[#,{k}]&/@#]]&/@IntegerPartitions[n,{k}])],{k,Sqrt[n]}],{n,30}]