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%I #12 Feb 08 2020 13:33:39
%S 1,1,1,4,4,7,16,19,28,40,82,94,145,190,274,463,580,802,1096,1486,1948
%N Number of strict solid partitions of n.
%C A strict solid partition is an infinite three-dimensional array of distinct positive integers (and any number of zeros) summing to n such that all one-dimensional sections are strictly decreasing until they become all zeros.
%e The a(1) = 1 through a(6) = 16 strict solid partitions, represented as chains of chains of integer partitions:
%e ((1)) ((2)) ((3)) ((4)) ((5)) ((6))
%e ((21)) ((31)) ((32)) ((42))
%e ((2)(1)) ((3)(1)) ((41)) ((51))
%e ((2))((1)) ((3))((1)) ((3)(2)) ((321))
%e ((4)(1)) ((4)(2))
%e ((3))((2)) ((5)(1))
%e ((4))((1)) ((31)(2))
%e ((32)(1))
%e ((4))((2))
%e ((5))((1))
%e ((31))((2))
%e ((3)(2)(1))
%e ((32))((1))
%e ((3)(1))((2))
%e ((3)(2))((1))
%e ((3))((2))((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 ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
%t strplptns[n_]:=Join@@Table[Select[ptnplane[Times@@Prime/@y],And[And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}]
%t Table[Length[Join@@Table[Select[Tuples[strplptns/@y],And[UnsameQ@@Flatten[#],And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#,{n,n}]&/@#)])]&],{y,IntegerPartitions[n]}]],{n,10}]
%Y Cf. A000219, A000293 (solid partitions), A000334, A001970, A002974, A114736, A117433 (strict plane partitions), A321662, A323657.
%K nonn,more
%O 0,4
%A _Gus Wiseman_, Jan 22 2019
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