A097516 a(n) counts the solid partitions of n that are symmetric under all of the operations mirroring (F), rotation (T) and 4-D rotation (L).

1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 3, 1, 1, 0, 4, 2, 2, 0, 4, 2, 3, 1, 4, 2, 3, 1, 6, 3, 3, 1, 7, 5, 5, 2, 7, 5, 6, 4, 7, 5, 6, 4, 9, 6, 8, 5, 10, 8, 12, 9, 11, 8, 13, 12, 13, 11, 13, 12, 15, 14, 17, 15, 16, 18, 22, 21, 18, 19, 23, 25, 20, 23, 27, 28, 22, 26, 34, 37, 26, 32, 39, 47, 31, 40

Offset: 1

Author

Wouter Meeussen, Sep 19 2004

Keywords

• nonn

Examples

			The totally symmetric solid partitions up to n=15 are:
[{{1}}]
[{{2,1}, {1}}, {{1}}]
[{{3,1,1}, {1}, {1}}, {{1}}, {{1}}]
[{{2,2}, {2,1}}, {{2,1}, {1}}]
[{{4,1,1,1}, {1}, {1}, {1}}, {{1}}, {{1}}, {{1}}]
[{{3,2,1}, {2,1}, {1}}, {{2,1}, {1}}, {{1}}] and
[{{2,2}, {2,2}}, {{2,2}, {2,1}}]
A list of weakly decreasing 4-tuples is enough to specify a totally symmetric solid partition.  First, think of a solid partition as a set of points in a 4-dimensional integral lattice in the standard way.  (Here I take the point (1, 1, 1, 1)—rather than (0, 0, 0, 0)—to represent the sole partition of 1.  Thus, all points have coordinates which are strictly positive.)
Now, associate to a weakly decreasing 4-tuples the smallest totally symmetric solid partition containing each of the listed 4-tuples as points.  For instance, the partition, call it p, which is represented by the list:
{(3, 1, 1, 1), (2, 2, 2, 1)}
is found by first noting that all points of the form (a, b, c, d) where a<=3, b<=1, c<=1, d<=1 (i.e the points (2, 1, 1, 1) and (1, 1, 1, 1)) must be points of p.  Similarly, all points (x, y, z, w) with x<=2, y<=2, z<=2, w<=1, must be points of p.  Furthermore all permutations of the coordinates of a point of p must also give a point of p by symmetry: E.g., since (2, 2, 1, 1) is a point of p, so are (2, 1, 2, 1), (2, 1, 1, 2), (1, 2, 2, 1), etc.  If we count all the points of p, we see p partitions 19.
Using this notation, we may represent the 5 totally symmetric solid partitions of 62 as:
1.  {(3, 3, 2, 1), (2, 2, 2, 2)}
2.  {(5, 1, 1, 1), (3, 3, 1, 1), (3, 2, 2, 2)}
3.  {(9, 1, 1, 1), (3, 3, 1, 1), (2, 2, 2, 2)}
4.  {(6, 1, 1, 1), (3, 3, 1, 1), (3, 2, 2, 1), (2, 2, 2, 2)}
5.  {(6, 1, 1, 1), (4, 2, 1, 1), (3, 3, 1, 1), (2, 2, 2, 2)}
		

Crossrefs

Cf. A000219, A096573, A096575, A096577, A097507.

Extensions

a(16)-a(32) from Suresh Govindarajan, Jun 07 2013

More terms and example text added by Graham H. Hawkes, Dec 24 2013