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%I #21 Oct 17 2014 22:14:31
%S 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,
%T 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,
%U 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
%N 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).
%e The totally symmetric solid partitions up to n=15 are:
%e [{{1}}]
%e [{{2,1}, {1}}, {{1}}]
%e [{{3,1,1}, {1}, {1}}, {{1}}, {{1}}]
%e [{{2,2}, {2,1}}, {{2,1}, {1}}]
%e [{{4,1,1,1}, {1}, {1}, {1}}, {{1}}, {{1}}, {{1}}]
%e [{{3,2,1}, {2,1}, {1}}, {{2,1}, {1}}, {{1}}] and
%e [{{2,2}, {2,2}}, {{2,2}, {2,1}}]
%e 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.)
%e 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:
%e {(3, 1, 1, 1), (2, 2, 2, 1)}
%e 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.
%e Using this notation, we may represent the 5 totally symmetric solid partitions of 62 as:
%e 1. {(3, 3, 2, 1), (2, 2, 2, 2)}
%e 2. {(5, 1, 1, 1), (3, 3, 1, 1), (3, 2, 2, 2)}
%e 3. {(9, 1, 1, 1), (3, 3, 1, 1), (2, 2, 2, 2)}
%e 4. {(6, 1, 1, 1), (3, 3, 1, 1), (3, 2, 2, 1), (2, 2, 2, 2)}
%e 5. {(6, 1, 1, 1), (4, 2, 1, 1), (3, 3, 1, 1), (2, 2, 2, 2)}
%Y Cf. A000219, A096573, A096575, A096577, A097507.
%K nonn
%O 1,15
%A _Wouter Meeussen_, Sep 19 2004
%E a(16)-a(32) from _Suresh Govindarajan_, Jun 07 2013
%E More terms and example text added by _Graham H. Hawkes_, Dec 24 2013
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