A323657 Number of strict solid partitions of n.

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

Offset

0,4

Comments

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.

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 * A323658 A323659 A323660

Keyword

nonn,more

Author

Gus Wiseman, Jan 22 2019

Status

approved