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%I A234859 #12 Jan 18 2024 11:01:27
%S A234859 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,
%T A234859 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,
%U A234859 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
%N A234859 Number of totally symmetric 5-dimensional partitions of n.
%C A234859 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.
%C A234859 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.
%C A234859 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.
%C A234859 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.
%K A234859 nonn
%O A234859 1,16
%A A234859 _Graham H. Hawkes_, Jan 01 2014

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