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%I #5 Jan 12 2019 20:43:11
%S 1,1,1,2,1,4,1,2,2,4,1,6,1,4,4,3,1,6,1,6,4,4,1,12,2,4,2,6,1,12,1,2,4,
%T 4,4,18,1,4,4,12,1,12,1,6,6,4,1,10,2,6,4,6,1,12,4,12,4,4,1,36,1,4,6,4,
%U 4,12,1,6,4,12,1,20,1,4,6,6,4,12,1,10,3,4
%N Number of ways to fill a matrix with the parts of the integer partition with Heinz number n.
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%F a(n) = A008480(n) * A000005(A001222(n)).
%e The a(24) = 12 matrices whose entries are (2,1,1,1):
%e [1 1 1 2] [1 1 2 1] [1 2 1 1] [2 1 1 1]
%e .
%e [1 1] [1 1] [1 2] [2 1]
%e [1 2] [2 1] [1 1] [1 1]
%e .
%e [1] [1] [1] [2]
%e [1] [1] [2] [1]
%e [1] [2] [1] [1]
%e [2] [1] [1] [1]
%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 Array[Length[ptnmats[#]]&,100]
%Y Positions of 1's are one and prime numbers A008578.
%Y Positions of 2's are primes to prime powers A053810.
%Y Cf. A000005, A001222, A008480, A056239, A063989, A112798, A120733.
%Y Cf. A323295, A323305, A323307, A323351.
%K nonn
%O 1,4
%A _Gus Wiseman_, Jan 12 2019
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