OFFSET
1,2
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Suresh Govindarajan, Table of n, a(n) for n = 1..40
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy], DOI
S. Balakrishnan, S. Govindarajan and N. S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231 [cond-mat.stat-mech], 2011.
S. P. Naveen, On The Asymptotics of Some Counting Problems in Physics, Thesis, Bachelor of Technology, Department of Physics, Indian Institute of Technology, Madras, May 2011.
EXAMPLE
From Gus Wiseman, Jan 23 2019: (Start)
The a(1) = 1 through a(3) = 15 four-dimensional partitions, represented as chains of chains of chains of integer partitions:
(((1))) (((2))) (((3)))
(((11))) (((21)))
(((1)(1))) (((111)))
(((1))((1))) (((2)(1)))
(((1)))(((1))) (((11)(1)))
(((2))((1)))
(((1)(1)(1)))
(((11))((1)))
(((2)))(((1)))
(((1)(1))((1)))
(((11)))(((1)))
(((1))((1))((1)))
(((1)(1)))(((1)))
(((1))((1)))(((1)))
(((1)))(((1)))(((1)))
(End)
MATHEMATICA
trans[x_]:=If[x=={}, {}, Transpose[x]];
levptns[n_, k_]:=If[k==1, IntegerPartitions[n], Join@@Table[Select[Tuples[levptns[#, k-1]&/@y], And@@(GreaterEqual@@@trans[Flatten/@(PadRight[#, ConstantArray[n, k-1]]&/@#)])&], {y, IntegerPartitions[n]}]];
Table[Length[levptns[n, 4]], {n, 8}] (* Gus Wiseman, Jan 24 2019 *)
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Sean A. Irvine, Nov 14 2010
STATUS
approved