A Note on Modular Partitions and Necklaces

@inproceedings{Sloane2014ANO,
  title={A Note on Modular Partitions and Necklaces},
  author={N. J. A. Sloane},
  year={2014}
}
Following Jens Voß [9], let T (n, k) be the number of k-tuples u = (u1, u2, . . . , uk) with 0 ≤ u1 ≤ u2 ≤ · · · ≤ uk ≤ n − 1 such that ∑ j uj ≡ 0 mod n. Stated another way, T (n, k) is the number of ways to write 0 as a sum of k elements of Z/nZ. Voß calls u a modular partition of n into k parts. He computed the numbers T (n, k) for n + k ≤ 20, and part of his table is shown here (the rows correspond to n = 0, 1, 2, . . . , 10 and the columns to k = 0, 1, 2, . . .):