A234968 Number of totally symmetric partitions of n of any dimension.

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

Offset: 2

Author

Graham H. Hawkes, Jan 02 2014

Keywords

• nonn
• nice

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.

Examples

			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.
		

Links

• Graham H. Hawkes, Table of n, a(n) for n = 2..90
• Graham H. Hawkes, Table of TS FD for dim 1...7

Crossrefs

The number of TS FD of dimensions 2, 3, and 4 are given by sequences A000700, A048141, and A097516 respectively.