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Number of integer partitions of n with the same product of parts as their conjugate.
20

%I #9 Jun 27 2020 04:36:35

%S 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,

%T 39,28,40,53,45,52,59,71,61,73,97,102,95,112,131,137,148,140,166,199,

%U 181,238,251,255,289,339,344,381,398,422,464,541,555,628,677,732

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

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

%C The Heinz numbers of these partitions are given by A325040.

%e The a(8) = 6 through a(15) = 6 integer partitions:

%e (44) (333) (4321) (641) (4422) (4432) (6431)

%e (332) (51111) (52111) (4331) (53211) (6421) (8411)

%e (431) (322211) (621111) (53311) (54221)

%e (2222) (611111) (432211) (433211)

%e (3221) (7111111) (632111)

%e (4211) (7211111)

%e (42221111)

%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];

%t Table[Length[Select[IntegerPartitions[n],Times@@#==Times@@conj[#]&]],{n,0,30}]

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

%Y Cf. A325040, A325041, A325042, A325045.

%K nonn

%O 0,9

%A _Gus Wiseman_, Mar 25 2019

%E More terms from _Jinyuan Wang_, Jun 27 2020