  We study M(n,k,r), the number of orbits of {(a_1,...,a_k)\in Z_n^k |
a_1+...+a_k = r (mod n)} under the action of S_k. Equivalently, M(n,k,r) sums
the partition numbers of an arithmetic sequence: M(n,k,r) = sum_{t \geq 0}
p(n-1,k,r+nt), where p(a,b,t) denotes the number of partitions of t into at
most b parts, each of which is at most a. We derive closed formulas and various
identities for such arithmetic partition sums.
  These results have already appeared in Elashvili/Jibladze/Pataraia,
Combinatorics of necklaces and "Hermite reciprocity", J. Alg. Combin. 10 (1999)
173-188, and the main result was also published by Von Sterneck in Sitzber.
Akad. Wiss. Wien. Math. Naturw. Class. 111 (1902), 1567-1601 (see Lemma 2 and
references in math.NT/9909121). Thanks to Don Zagier and Robin Chapman for
bringing these references to our attention.
