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A323657 Number of strict solid partitions of n. 3
1, 1, 1, 4, 4, 7, 16, 19, 28, 40, 82, 94, 145, 190, 274, 463, 580, 802, 1096, 1486, 1948 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A strict solid partition is an infinite three-dimensional array of distinct positive integers (and any amount of zeros) summing to n such that all one-dimensional sections are strictly decreasing until they become all zeros.

LINKS

Table of n, a(n) for n=0..20.

EXAMPLE

The a(1) = 1 through a(6) = 16 strict solid partitions, represented as chains of chains of integer partitions:

  ((1))  ((2))  ((3))       ((4))       ((5))       ((6))

                ((21))      ((31))      ((32))      ((42))

                ((2)(1))    ((3)(1))    ((41))      ((51))

                ((2))((1))  ((3))((1))  ((3)(2))    ((321))

                                        ((4)(1))    ((4)(2))

                                        ((3))((2))  ((5)(1))

                                        ((4))((1))  ((31)(2))

                                                    ((32)(1))

                                                    ((4))((2))

                                                    ((5))((1))

                                                    ((31))((2))

                                                    ((3)(2)(1))

                                                    ((32))((1))

                                                    ((3)(1))((2))

                                                    ((3)(2))((1))

                                                    ((3))((2))((1))

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];

ptnplane[n_]:=Union[Map[Reverse@*primeMS, Join@@Permutations/@facs[n], {2}]];

strplptns[n_]:=Join@@Table[Select[ptnplane[Times@@Prime/@y], And[And@@GreaterEqual@@@#, And@@(GreaterEqual@@@Transpose[PadRight[#]])]&], {y, Select[IntegerPartitions[n], UnsameQ@@#&]}]

Table[Length[Join@@Table[Select[Tuples[strplptns/@y], And[UnsameQ@@Flatten[#], And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#, {n, n}]&/@#)])]&], {y, IntegerPartitions[n]}]], {n, 10}]

CROSSREFS

Cf. A000219, A000293 (solid partitions), A000334, A001970, A002974, A114736, A117433 (strict plane partitions), A321662, A323657.

Sequence in context: A284640 A036605 A183541 * A238389 A115292 A202676

Adjacent sequences:  A323654 A323655 A323656 * A323659 A323660 A323661

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jan 22 2019

STATUS

approved

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Last modified June 16 14:56 EDT 2019. Contains 324152 sequences. (Running on oeis4.)