Veit Elser writes:
It's easy to show
S(k m, k p) >= S(m, p).
Proof: I prefer the symmetrical quantity U(m, p) = p S(m, p). [etc.]
It's actually easier to see this fact if one uses S instead of U and thinks in terms of actual muffins: If you've got km muffins and kp people, divide the first m muffins among the first p people, the next m muffins among the next p people, etc. If in each batch of p people nobody gets a piece of size smaller than S(m,p), then none of the mp people gets a piece of size smaller than S(m,p), so we've found a way to divide km mufins among kp people so that nobody gets a piece of size smaller than S(m,p). That is, S(km,kp) is at least as large as S(m,p). I suspect that S(km,kp) sometimes is strictly larger than S(m,p); what do the rest of you think? Jim Propp