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%I #4 Jan 13 2019 08:50:28
%S 1,1,1,2,1,0,1,2,2,0,1,0,1,0,0,3,1,0,1,0,0,0,1,0,2,0,2,0,1,0,1,2,0,0,
%T 0,2,1,0,0,0,1,0,1,0,0,0,1,0,2,0,0,0,1,0,0,0,0,0,1,0,1,0,0,4,0,0,1,0,
%U 0,0,1,0,1,0,0,0,0,0,1,0,3,0,1,0,0,0,0
%N Number of ways to arrange the parts of the integer partition with Heinz number n into a matrix with equal row-sums and equal column-sums.
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%e The a(900) = 12 matrix-arrangements of (3,3,2,2,1,1):
%e [1 2 3] [1 3 2] [2 1 3] [2 3 1] [3 1 2] [3 2 1]
%e [3 2 1] [3 1 2] [2 3 1] [2 1 3] [1 3 2] [1 2 3]
%e .
%e [1 3] [1 3] [2 2] [2 2] [3 1] [3 1]
%e [2 2] [3 1] [1 3] [3 1] [1 3] [2 2]
%e [3 1] [2 2] [3 1] [1 3] [2 2] [1 3]
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
%t Table[Length[Select[ptnmats[n],And[SameQ@@Total/@#,SameQ@@Total/@Transpose[#]]&]],{n,100}]
%Y Positions of zeros are a superset of A106543.
%Y Cf. A000005, A001222, A006052, A007016, A008480, A056239, A112798, A120733, A319056, A321719, A321721.
%Y Cf. A323300, A323303, A323305, A323306, A323347, A323349.
%K nonn
%O 1,4
%A _Gus Wiseman_, Jan 13 2019
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