This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A323657 #22 Dec 20 2024 16:18:57
%S A323657 1,1,1,4,4,7,16,19,28,40,82,94,145,190,274,463,580,802,1096,1486,1948,
%T A323657 3148,3811,5314,6922,9394,11971,16156,23044,28966,38368,50002,65116,
%U A323657 83872,108706,137917,192070,236242,308698,390772,506935,633982,817324,1018090
%N A323657 Number of strict solid partitions of n.
%C A323657 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.
%H A323657 Alois P. Heinz, <a href="/A323657/b323657.txt">Table of n, a(n) for n = 0..740</a> (first 401 terms from John Tyler Rascoe)
%F A323657 a(n) = Sum_{k=1..n} A008289(n,k)*A207542(k) for n > 0. - _John Tyler Rascoe_, Dec 19 2024
%e A323657 The a(1) = 1 through a(6) = 16 strict solid partitions, represented as chains of chains of integer partitions:
%e A323657 ((1)) ((2)) ((3)) ((4)) ((5)) ((6))
%e A323657 ((21)) ((31)) ((32)) ((42))
%e A323657 ((2)(1)) ((3)(1)) ((41)) ((51))
%e A323657 ((2))((1)) ((3))((1)) ((3)(2)) ((321))
%e A323657 ((4)(1)) ((4)(2))
%e A323657 ((3))((2)) ((5)(1))
%e A323657 ((4))((1)) ((31)(2))
%e A323657 ((32)(1))
%e A323657 ((4))((2))
%e A323657 ((5))((1))
%e A323657 ((31))((2))
%e A323657 ((3)(2)(1))
%e A323657 ((32))((1))
%e A323657 ((3)(1))((2))
%e A323657 ((3)(2))((1))
%e A323657 ((3))((2))((1))
%t A323657 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A323657 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A323657 ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
%t A323657 strplptns[n_]:=Join@@Table[Select[ptnplane[Times@@Prime/@y],And[And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}]
%t A323657 Table[Length[Join@@Table[Select[Tuples[strplptns/@y],And[UnsameQ@@Flatten[#],And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#,{n,n}]&/@#)])]&],{y,IntegerPartitions[n]}]],{n,10}]
%Y A323657 Cf. A000219, A000293 (solid partitions), A000334, A001970, A002974, A008289, A114736, A117433 (strict plane partitions), A207542, A321662, A323657.
%K A323657 nonn
%O A323657 0,4
%A A323657 _Gus Wiseman_, Jan 22 2019
%E A323657 a(21) onwards from _John Tyler Rascoe_, Dec 19 2024