A002974 Number of restricted solid partitions of n.

1, 1, 4, 7, 11, 20, 35, 59, 99, 165, 270, 443, 723, 1161, 1861, 2961, 4654, 7279, 11317, 17476, 26879, 41132, 62601, 94878, 143172, 215115, 321995, 480216, 713655, 1057192

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• more

Comments

Definition, based on Math. Review MR0297583: By a solid partition of n is meant a 3-dimensional arrangement of positive integers N(x,y,z) satisfying the conditions (i) the integer N(x,y,z) is located at the point with Cartesian coordinates (x,y,z); N(x,y,z) is defined only for certain integers x,y,z >= 0, and (ii) if N(x,y,z) is defined and 0 <= x' <= x, 0 <= y' <= y, 0 <= z' <= z then N(x,y,z) is defined and N(x',y',z') <= N(x,y,z). A solid partition is said to correspond to an (ordinary) partition of n=n_1+n_2+...+n_t, n_k>0, if there is a one-to-one correspondence between the summands n_k and the points (x_k,y_k,z_k) for which N is defined so that n_k=N(x_k,y_k,z_k). Finally, a restricted solid partition is a solid partition such that x'<=x, y'<=y, z'<=z and N(x',y',z')=N(x,y,z) implies x'=x, y'=y, z'=z.

Alternatively, a restricted solid partition is an infinite three-dimensional array of nonnegative integers summing to n such that all one-dimensional sections are strictly decreasing until they become all zeros. - Gus Wiseman, Jan 22 2019

Examples

			From _Gus Wiseman_, Jan 22 2019: (Start)
The a(1) = 1 through a(6) = 20 restricted 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))  ((21)(1))      ((3)(2))        ((321))
                            ((3))((1))     ((4)(1))        ((4)(2))
                            ((21))((1))    ((31)(1))       ((5)(1))
                            ((2)(1))((1))  ((3))((2))      ((31)(2))
                                           ((4))((1))      ((32)(1))
                                           ((31))((1))     ((41)(1))
                                           ((3)(1))((1))   ((4))((2))
                                           ((21)(1))((1))  ((5))((1))
                                                           ((31))((2))
                                                           ((3)(2)(1))
                                                           ((32))((1))
                                                           ((41))((1))
                                                           ((3)(1))((2))
                                                           ((3)(2))((1))
                                                           ((4)(1))((1))
                                                           ((31)(1))((1))
                                                           ((3))((2))((1))
(End)
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• H. Gupta, Restricted solid partitions, J. Combin. Theory, A 13 (1972), 140-144.

Crossrefs

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

Programs

Mathematica

srcplptns[n_]:=Join@@Table[Select[Tuples[IntegerPartitions/@ptn],And[And@@(GreaterEqual@@@Transpose[PadRight[#]]),And@@Greater@@@#,And@@(Greater@@@DeleteCases[Transpose[PadRight[#]],0,{2}])]&],{ptn,IntegerPartitions[n]}];
srcsolids[n_]:=Join@@Table[Select[Tuples[srcplptns/@y],And[And@@(GreaterEqual@@@Transpose[Join@@@(PadRight[#,{n,n}]&/@#)]),And@@(Greater@@@DeleteCases[Transpose[Join@@@(PadRight[#,{n,n}]&/@#)],0,{2}])]&],{y,IntegerPartitions[n]}]
Table[Length[srcsolids[n]],{n,10}] (* Gus Wiseman, Jan 23 2019 *)

Extensions

More terms from Sean A. Irvine, Dec 15 2014