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A234968 Number of totally symmetric partitions of n of any dimension. 1
1, 2, 3, 3, 3, 5, 5, 5, 6, 7, 5, 9, 6, 9, 13, 11, 7, 16, 14, 14, 16, 19, 14, 23, 24, 21, 27, 32, 21, 39, 39, 32, 38, 51, 45, 56, 60, 51, 62, 87, 61, 82, 101, 83, 98, 129, 104, 120, 152, 137, 145, 196, 157, 178, 248, 207, 209, 293, 248, 275, 353, 310, 325, 441, 388, 389, 528, 471, 463, 656, 573, 567, 766, 696, 691, 934 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
COMMENTS
a(n) is the sum over d from 1 to infinity of the number of totally symmetric d-dimensional Ferrers diagrams with n nodes.
A d-dimensional Ferrers diagram is totally symmetric if and only if whenever X=(x1,x2,...,xd) is a node, then so are all nodes which can be specified by permuting the coordinates of X.
Since a(1)=oo, the sequence above begins on n=2. All other terms are finite.
LINKS
EXAMPLE
a(1)=oo because for each dimension, d, the trivial Ferrers diagram given by the single node (1,1,1,...,1) is a totally symmetric d-dimensional partition of 1.
For n > 2, a(n) < oo. This means that for n > 2, there are at most a finite number of dimensions, d, for which the number of totally symmetric d-dimensional partitions of n is nonzero (and that for any dimension, d, there are at most a finite number of totally symmetric d-dimensional partitions of n).
a(2)=1. Indeed the only totally symmetric partition of 2 occurs in dimension 1. The corresponding 1-dimensional totally symmetric Ferrers diagram (TS FD) is given by the following two nodes (specified by the 1-dimensional coordinates): (2) and (1).
a(8)=5.
There is one 1-dimensional TS FD of 8:
{(8),(7),(6),(5),(4),(3),(2),(1)}
There are two 2-dimensional TS FD of 8:
{(3,2),(2,3),(3,1),(2,2),(1,3),(2,1),(1,2),(1,1) and
{(4,1),(1,4),(3,1),(2,2),(1,3),(2,1),(1,2),(1,1)}
There is one 3-dimensional TS FD of 8:
{(2,2,2),(2,2,1),(2,1,2),(1,2,2),(2,1,1),(1,2,1),(1,1,2),(1,1,1)}
There is one 7-dimensional TS FD of 8:
{(2,1,1,1,1,1,1),(1,2,1,1,1,1,1),(1,1,2,1,1,1,1),(1,1,1,2,1,1,1),(1,1,1,1,2,1,1),(1,1,1,1,1,2,1),(1,1,1,1,1,1,2),(1,1,1,1,1,1,1)}
There are no TS FD of 8 of any other dimension. Hence a(8)=1+2+1+1=5.
a(72)=573
The TS FD of 72 are:
Dim 1: 1
Dim 2: 471
Dim 3: 85
Dim 4: 11
Dim 5: 3
Dim 6: 1
Dim 71: 1
(For n > 1) there is always exactly 1 TS FD of dimension 1 and 1 TS FD of dimension n-1. If n > 2, these two dimensions are not equal, so there must be at least two TS FD. Hence a(n) >= 2 for n > 2.
CROSSREFS
The number of TS FD of dimensions 2, 3, and 4 are given by sequences A000700, A048141, and A097516 respectively.
Sequence in context: A087731 A308172 A227559 * A141784 A234337 A216391
KEYWORD
nonn,nice
AUTHOR
Graham H. Hawkes, Jan 02 2014
STATUS
approved

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