A234859 Number of totally symmetric 5-dimensional partitions of n.

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

Offset: 1

Author

Graham H. Hawkes, Jan 01 2014

Keywords

• nonn

Comments

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.