A002974 Number of restricted solid partitions of n. (Formerly M3304)

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

1,3

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

References

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

Links

Table of n, a(n) for n=1..30.

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

Example

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)

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 *)

Crossrefs

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

Sequence in context: A228079 A083839 A091176 * A130625 A104102 A074705

Adjacent sequences: A002971 A002972 A002973 * A002975 A002976 A002977

Keyword

nonn,more

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Dec 15 2014

Status

approved