A325039 Number of integer partitions of n with the same product of parts as their conjugate.

1, 1, 0, 1, 1, 1, 1, 1, 6, 2, 2, 4, 3, 5, 7, 6, 5, 7, 9, 10, 11, 18, 16, 19, 19, 16, 20, 20, 28, 39, 28, 40, 53, 45, 52, 59, 71, 61, 73, 97, 102, 95, 112, 131, 137, 148, 140, 166, 199, 181, 238, 251, 255, 289, 339, 344, 381, 398, 422, 464, 541, 555, 628, 677, 732

Offset

0,9

Comments

For example, the partition (6,4,1) with product 24 has conjugate (3,2,2,2,1,1) with product also 24.

The Heinz numbers of these partitions are given by A325040.

Links

Table of n, a(n) for n=0..64.

Example

The a(8) = 6 through a(15) = 6 integer partitions:
  (44)    (333)    (4321)   (641)     (4422)    (4432)     (6431)
  (332)   (51111)  (52111)  (4331)    (53211)   (6421)     (8411)
  (431)                     (322211)  (621111)  (53311)    (54221)
  (2222)                    (611111)            (432211)   (433211)
  (3221)                                        (7111111)  (632111)
  (4211)                                                   (7211111)
                                                           (42221111)

Mathematica

conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Length[Select[IntegerPartitions[n], Times@@#==Times@@conj[#]&]], {n, 0, 30}]

Crossrefs

Cf. A001055, A064573, A122111, A296150, A318950, A319000, A320322, A321649.

Cf. A325040, A325041, A325042, A325045.

Sequence in context: A276801 A083286 A247818 * A172439 A169684 A259838

Adjacent sequences: A325036 A325037 A325038 * A325040 A325041 A325042

Keyword

nonn

Author

Gus Wiseman, Mar 25 2019

Extensions

More terms from Jinyuan Wang, Jun 27 2020

Status

approved