A323657 - OEIS

%I #22 Dec 20 2024 16:18:57

%S 1,1,1,4,4,7,16,19,28,40,82,94,145,190,274,463,580,802,1096,1486,1948,

%T 3148,3811,5314,6922,9394,11971,16156,23044,28966,38368,50002,65116,

%U 83872,108706,137917,192070,236242,308698,390772,506935,633982,817324,1018090

%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.

%H 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 a(n) = Sum_{k=1..n} A008289(n,k)*A207542(k) for n > 0. - _John Tyler Rascoe_, Dec 19 2024

%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, A008289, A114736, A117433 (strict plane partitions), A207542, A321662, A323657.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jan 22 2019

%E a(21) onwards from _John Tyler Rascoe_, Dec 19 2024