trying to confirm scott's data ...
Here are some example partitions m/p. These are given with integral pieces, which one may normalize if desired. I believe these are all optimal.
7/5: pieces 5,7,8: 5.5.5, 7.8*6; 7.7.7*2, 8.8.5*3 S = 1/3, R_U = 8/5 (note: 7.8*6 = 6 muffins split 7.8, 8.8.5*3 = 3 people with pieces 8.8.5)
confirmed. i can prove that T(5, 7) = 1/21 , and an optimal partition must have a part >= 8/105 . this seems to be the smallest case where one of the smaller fractions must be split into 3 parts.
10/7: pieces 7..13: 7.7.7, 13.8*3, 12.9*3, 11.10*3; 7.11.12*3, 3.9.13*3, 10.10.10 S = 1/3, R_U = 13/7
i can prove that T(7, 10) = 1/30 , which confirms your value of S . however, i can improve the ratio to 23/14 with the partition 3 * [1/30 + 2 * 23/420] + 3 * [2 * 19/420 + 11/210] + [3 * 1/21] <---> [3 * 1/30] + 6 * [19/420 + 23/420] + 3 * [1/21 + 11/210] . this is probably the smallest max/min ratio.
11/10: pieces 10..19: 10.10.10*2, 11.19*2, 12.18*2, 13.17*2, 14.16*2, 15.15; 10.10.13*2, 10.11.12*2, 14.19*2, 15.18*2, 16.17*2 S = 1/3, R_U = 19/10
i think i can prove that T(10, 11) = 7/220 , which improves your value of S . in any event, that's a lower bound, obtained by the partition 2 * [2 * 7/220 + 2/55] + 4 * [9/220 + 13/220] + 2 * [1/22 + 3/55] + 2 * [2 * 1/20] <---> 4 * [7/220 + 13/220] + 2 * [2/55 + 3/55] + 4 * [9/220 + 1/20] + [2 * 1/22] .
12/7: pieces 3..4: 3.4*12; 3.3.3.3*3, 4.4.4*4 S = 3/7, R_U = 4/3
confirmed. i can prove that T(7, 12) = 1/28 , and the optimal partition is unique. this generalizes to T(2n+1, n(n+1)) = 1/((n+1)(2n+1)) , with unique optimal partition n * [(n+1) * 1/((n+1)(2n+1))] + (n+1) * [n * 1/(n(2n+1))] <---> n(n+1) * [1/((n+1)(2n+1)) + 1/(n(2n+1))] .
9/5: pieces 4..6: 5.5*3, 4.6*6; 6.6.6*2, 4.4.5.5*3 S = 2/5, R_U = 3/2
confirmed. i can prove that T(5, 9) = 2/45 , and an optimal partition must have a part >= 1/15 , which shows that your ratio is best possible.
16/9: pieces 11.16: 11.16*12, 12.15*2, 13.14*2; 16.16.16*4, 11.11.11.15*2, 11.11.12.14*2, 11.11.13.13 S = 11/27, R_U = 16/11
confirmed. i can prove that T(9, 16) = 11/432 , and an optimal partition must have a part >= 1/27 , which shows that your ratio is best possible.
23/13: pieces 21..31: 21.31*12, 22.30*6, 23.29*4, 26.26; 31.31.30*6, 21.21.21.29*4, 22.22.22.26*2, 23.23.23.23 S = 21/52, R_U = 31/21
ouch! i can't manage anything with this one yet ...
7/4: pieces 5..7: 5.7*6, 6.6; 7.7.7*2, 5.5.5.6*2 S = 5/12, R_U = 7/5
confirmed. T(4, 7) = 5/84 , and the optimal partition is unique.
17/9: pieces 10..17: 10.17*6, 11.16, 12.15, 13.14*9; 17.17.17*2, 10.12.13.16, 10.13.13.15, 11.13.13.14, 10.13.14.14*4 S = 10/27, R_U = 17/10
confirmed. i can prove that T(9, 17) = 10/459 , and an optimal partition must have a part >= 1/27 , which shows that your ratio is best possible. mike