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

2,2

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

Graham H. Hawkes, Table of n, a(n) for n = 2..90

Graham H. Hawkes, Table of TS FD for dim 1...7

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

Adjacent sequences: A234965 A234966 A234967 * A234969 A234970 A234971

Keyword

nonn,nice

Author

Graham H. Hawkes, Jan 02 2014

Status

approved