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Number of restricted solid partitions of n.
(Formerly M3304)
3

%I M3304 #29 Jan 23 2019 19:59:42

%S 1,1,4,7,11,20,35,59,99,165,270,443,723,1161,1861,2961,4654,7279,

%T 11317,17476,26879,41132,62601,94878,143172,215115,321995,480216,

%U 713655,1057192

%N Number of restricted solid partitions of n.

%C 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.

%C 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

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

%H H. Gupta, <a href="http://dx.doi.org/10.1016/0097-3165(72)90018-0">Restricted solid partitions</a>, J. Combin. Theory, A 13 (1972), 140-144.

%e From _Gus Wiseman_, Jan 22 2019: (Start)

%e The a(1) = 1 through a(6) = 20 restricted solid partitions, represented as chains of chains of integer partitions:

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

%e ((21)) ((31)) ((32)) ((42))

%e ((2)(1)) ((3)(1)) ((41)) ((51))

%e ((2))((1)) ((21)(1)) ((3)(2)) ((321))

%e ((3))((1)) ((4)(1)) ((4)(2))

%e ((21))((1)) ((31)(1)) ((5)(1))

%e ((2)(1))((1)) ((3))((2)) ((31)(2))

%e ((4))((1)) ((32)(1))

%e ((31))((1)) ((41)(1))

%e ((3)(1))((1)) ((4))((2))

%e ((21)(1))((1)) ((5))((1))

%e ((31))((2))

%e ((3)(2)(1))

%e ((32))((1))

%e ((41))((1))

%e ((3)(1))((2))

%e ((3)(2))((1))

%e ((4)(1))((1))

%e ((31)(1))((1))

%e ((3))((2))((1))

%e (End)

%t 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]}];

%t 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]}]

%t Table[Length[srcsolids[n]],{n,10}] (* _Gus Wiseman_, Jan 23 2019 *)

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

%K nonn,more

%O 1,3

%A _N. J. A. Sloane_

%E More terms from _Sean A. Irvine_, Dec 15 2014