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
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)}
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