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A002974 Number of restricted solid partitions of n.
(Formerly M3304)
3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 20 17:51 EDT 2019. Contains 324234 sequences. (Running on oeis4.)