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A234859
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Number of totally symmetric 5-dimensional partitions of n.
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0
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1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 4, 1, 0, 0, 0, 4, 1, 0, 0, 0, 5, 1, 0, 0, 0, 5, 1, 0, 0, 0, 7, 1, 0, 0, 0, 8, 2, 0, 0, 0, 9, 2, 0, 0, 0, 1, 3, 0, 0, 0, 11, 3, 0, 0, 0, 13, 3, 0, 0, 0, 15, 4, 0, 0, 0, 17, 5, 0, 0, 0, 20, 6, 0, 0, 0, 22, 7, 0, 0, 0, 24, 7, 0, 0, 0, 29, 9, 0, 0, 0, 32, 12, 0, 0, 0, 37, 14, 0, 0, 0, 41, 17, 0, 0, 0
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OFFSET
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1,16
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COMMENTS
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a(n) gives the number of 5-dimensional Ferrers diagrams that have the property that if the point X=(x1, x2, x3, x4, x5) appears in the diagram, then so do all the points specified by the permutations of the coordinates of X.
We can think of the points of a totally symmetric partition of n, say p, as occurring in classes, where two points are in the same class iff one point is a given by a permutation of the coordinates of the other.
Suppose p is a 5-dimensional totally symmetric partition of n. For any point of n, say x = (x1, x2, x3, x4, x5), then, because 5 is prime, 5 divides the number of distinct permutations of the coordinates of x unless x1 = x2 = x3 = x4 = x5 (in which case there is only 1 such distinct permutation). Therefore, the only classes of points in p which have a number of points not divisible by 5 are points of the form (x,x,x,x,x). Hence, the number of points in p is equal to m mod 5, where m is the number of diagonal points, or points of the form (x,x,x,x,x), in p.
If 0 < n < 32=2^5, then the number of diagonal points in any 5-dimensional partition of n must be less than 2 (and greater than 0)—therefore equal to 1. Thus, for n < 32, a(n) is nonzero only if n=1 mod 5. Further, if 0 < n < 243=3^5, then the number of diagonal points in any 5-dimensional partition of n must be less than 3, thus equal to 1 or 2. Thus for n < 243, a(n) is nonzero only if n=1 mod 5 or n=2 mod 5. Consequently for n=0, 3, or 4 mod 5, a(n)=0 in the first 125 terms given above. A similar pattern occurs in a sequence of totally symmetric d-dimensional partitions of n whenever d is prime.
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LINKS
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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