I decided that the one other value of m that it wouldn't be so bad to deal with is 8, so now I'll give lexically optimal partitions for U(8, n). As usual, for even values of n, the optimal partitions are obtained by taking those for U(4, n/2) and doubling all piece sizes and multiplicities. Here are the optimal partitions for odd n>=9. They all achieve equality in the usual upper bound U(m, n) <= min( n/([2n/m]+1), m - n/[2n/m] ), except for n=11, when the bound doesn't apply, and you can do better. For n=11, 17, the general constructions don't work, and I do them separately. I (usually) write parts with mixed fractions, and I use "-+" to mean that the indicated number of pieces occur both with the "-" sign and with the "+" sign. Also, I write e for either -1 or +1. (8, 11): 8/3 * 6, 11/3 * 6, 4 * 6, 13/3 * 6 (8, 17): 17/5 * 10, 19/5 * 2, 39/10 * 4, 4 * 2, 41/10 * 4, 21/5 * 2, 23/5 * 10 (8, 12k+e): [4 -+ 3/(3k+e)] * (6k+2e), [4 -+ 2/(2k-e)(3k+e)] * (4k-2e), [4 -+ 2/(2k-1+e)(2k-e)(3k+e)] * (2k-1+e), 4 * 2 (8, 12k+3e): [4 -+ 1/k] * 6k, 4 * 6, 8 * (6k-3+3e) (8, 12k+5e): [4 -+ 3/(3k+2e)] * (6k+4e), [4 -+ 2/(2k+e)(3k+2e)] * (4k+2e), [4 -+ 2/(2k-1-e)(2k+e)(3k+2e)] * (2k-1-e), 4 * 2 David P. Moulton