[math-fun] common refinements of partitions (was: muffins)
i'm glad to see that there is still some interest in the muffin/common refinement of partitions problem. some time ago, veit elser gave the upper bound T(m, n) <= 1/(3m) for m < n < 3m/2 , and noted that some of the values he found achieved this bound. i looked at this range more closely and found that i could improve the upper bound as follows. here t denotes a positive integer. if ((3t+1)/(3t))m < n <= ((3t)/(3t-1))m , then T(m, n) <= 1/(3n) . if ((3t+2)/(3t+1))m < n <= ((3t+1)/(3t))m , then T(m, n) <= (t+1)/(2n) - t/(2m) . if ((3t+3)/(3t+2))m < n < ((3t+2)/(3t+1))m , then T(m, n) <= (t+1)/(4m) - t/(4n) . if ((3t+3)/(3t+2))m = n , then T(m, n) <= 1/(3m) . (here, as in previous messages, i am using T(m, n) = maximum c such that there is a common refinement of 1/m + 1/m + ... + 1/m = 1 and of 1/n + 1/n + ... + 1/n = 1 , with all parts >= c .) the first three cases are all improvements over veit's bound. in the fourth case, where there is no improvement, equality is always attained by veit's construction for T(3t+2, 3t+3) , and by obvious extension to T((3t+2)k, (3t+3)k) . however, the above shows that this is the exception, rather than the rule. i find the first case interesting, because i have not seen any cases (with m <= n ) where T(m, n) is strictly less than 1/(3n) . if we allow parts of size 1/(3n) , we may split some of the 1/n 's into 3 parts, and that seems to allow a lot of flexibility. in fact, the only occurrences of T(m, n) = 1/(3n) i've seen are those from the first case above. so i conjecture that equality always holds in the first case. on the other hand, the bounds in the other cases cannot always be attained. they can be attained if n = m+1 , by veit's and moulton's constructions. they can also be attained if n = m+2 by similar constructions. however, if n >= m + 3 , there are cases that require more careful analysis and for which the upper bound can be lowered; the first such case is (13, 16) . when n = m + 3 , it seems that the upper bound can be attained in all cases except when m = 4 mod 9 ; in that case the exact value is T(9k+4, 9k+7) = (9k+5)/(3(9k+4)(9k+7)) for k >= 1 . below are some data i calculated, all by pencil and paper, so subject to a small possibility of error. obviously this amount of data starts to become unwieldy; corrections and/or confirmations are welcome! i believe i have a proof for each of these values, but may have given short shrift to some subcases i thought couldn't give optimal partitions. mike T(2, 3) = 1/6 T(2, 4) = 1/4 T(3, 4) = 1/12 T(2, 5) = 1/10 T(3, 5) = 1/12 T(4, 5) = 3/40 T(2, 6) = 1/6 T(3, 6) = 1/6 T(4, 6) = 1/12 T(5, 6) = 1/15 T(2, 7) = 1/14 T(3, 7) = 5/84 T(4, 7) = 5/84 T(5, 7) = 1/21 T(6, 7) = 1/21 T(2, 8) = 1/8 T(3, 8) = 1/18 T(4, 8) = 1/8 T(5, 8) = 1/20 T(6, 8) = 1/24 T(7, 8) = 5/112 T(2, 9) = 1/18 T(3, 9) = 1/9 T(4, 9) = 7/144 T(5, 9) = 2/45 T(6, 9) = 1/18 T(7, 9) = 5/126 T(8, 9) = 1/24 T(2, 10) = 1/10 T(3, 10) = 2/45 T(4, 10) = 1/20 T(5, 10) = 1/10 T(6, 10) = 1/24 T(7, 10) = 1/30 T(8, 10) = 3/80 T(9, 10) = 1/30 T(2, 11) = 1/22 T(3, 11) = 1/24 T(4, 11) = 9/220 T(5, 11) = 13/330 T(6, 11) = 7/198 T(7, 11) = 1/28 T(8, 11) = 1/33 T(9, 11) = 13/396 T(10, 11) = 7/220 T(2, 12) = 1/12 T(3, 12) = 1/12 T(4, 12) = 1/12 T(5, 12) = 1/30 T(6, 12) = 1/12 T(7, 12) = 1/28 T(8, 12) = 1/24 T(9, 12) = 1/36 T(10, 12) = 1/30 T(11, 12) = 1/33 T(2, 13) = 1/26 T(3, 13) = 11/312 T(4, 13) = 11/312 T(5, 13) = 1/30 T(6, 13) = 1/30 T(7, 13) = 8/273 T(8, 13) = 1/32 T(9, 13) = 1/39 T(10, 13) = 7/260 T(11, 13) = 1/39 T(12, 13) = 1/39 T(2, 14) = 1/14 T(3, 14) = 1/30 T(4, 14) = 1/28 T(5, 14) = 11/350 T(6, 14) = 5/168 T(7, 14) = 1/14 T(8, 14) = 5/168 T(9, 14) = 1/36 T(10, 14) = 1/42 T(11, 14) = 2/77 T(12, 14) = 1/42 T(13, 14) = 9/364 T(2, 15) = 1/30 T(3, 15) = 1/15 T(4, 15) = 13/420 T(5, 15) = 1/15 T(6, 15) = 1/30 T(7, 15) = 1/35 T(8, 15) = 1/40 T(9, 15) = 1/36 T(10, 15) = 1/30 T(11, 15) = 1/45 T(12, 15) = 1/40 T(13, 15) = 3/130 T(14, 15) = 1/42 T(2, 16) = 1/16 T(3, 16) = 7/240 T(4, 16) = 1/16 T(5, 16) = 1/35 T(6, 16) = 1/36 T(7, 16) = 3/112 T(8, 16) = 1/16 T(9, 16) = 11/432 T(10, 16) = 1/40 T(11, 16) = 1/48 T(12, 16) = 1/48 T(13, 16) = 7/312 T(14, 16) = 5/224 T(15, 16) = 1/48 T(2, 17) = 1/34 T(3, 17) = 1/36 T(4, 17) = 15/544 T(5, 17) = 13/510 T(6, 17) = 13/510 T(7, 17) = 11/476 T(8, 17) = 1/40 T(9, 17) = 10/459 T(10, 17) = 2/85 T(11, 17) = 1/44 T(12, 17) = 1/51 T(13, 17) = 9/442 T(14, 17) = 5/238 T(15, 17) = 7/340 T(16, 17) = 11/544 T(2, 18) = 1/18 T(3, 18) = 1/18 T(4, 18) = 1/36 T(5, 18) = 1/40 T(6, 18) = 1/18 T(7, 18) = 1/42 T(8, 18) = 7/288 T(9, 18) = 1/18 T(10, 18) = 1/45 T(11, 18) = 1/44 T(12, 18) = 1/36 T(13, 18) = 1/54 T(14, 18) = 5/252 T(15, 18) = 1/45 T(16, 18) = 1/48 T(17, 18) = 1/51 T(2, 19) = 1/38 T(3, 19) = 17/684 T(4, 19) = 17/684 T(5, 19) = 16/665 T(6, 19) = 1/42 T(7, 19) = 25/1064 T(8, 19) = 13/608 T(9, 19) = 1/45 T(10, 19) = 11/570 T(11, 19) = 9/418 T(12, 19) = 1/48 T(13, 19) = 1/57 T(14, 19) = 1/57 T(15, 19) = 7/380 T(16, 19) = 1/57 T(17, 19) = 1/57 T(18, 19) = 1/57 T(2, 20) = 1/20 T(3, 20) = 1/42 T(4, 20) = 1/20 T(5, 20) = 1/20 T(6, 20) = 1/45 T(7, 20) = 3/140 T(8, 20) = 1/40 T(9, 20) = 1/45 T(10, 20) = 1/20 T(11, 20) = 13/660 T(12, 20) = 1/48 T(13, 20) = 1/52 T(14, 20) = 1/60 T(15, 20) = 1/60 T(16, 20) = 3/160 T(17, 20) = 1/60 T(18, 20) = 1/60 T(19, 20) = 13/760 T(2, 21) = 1/42 T(3, 21) = 1/21 T(4, 21) = 19/840 T(5, 21) = 1/45 T(6, 21) = 1/42 T(7, 21) = 1/21 T(8, 21) = 1/48 T(9, 21) = 5/252 T(10, 21) = 1/50 T(11, 21) = 4/231 T(12, 21) = 5/252 T(13, 21) = 1/52 T(14, 21) = 1/42 T(15, 21) = 1/63 T(16, 21) = 11/672 T(17, 21) = 2/119 T(18, 21) = 1/63 T(19, 21) = 13/798 T(20, 21) = 1/60 T(2, 22) = 1/22 T(3, 22) = 5/231 T(4, 22) = 1/44 T(5, 22) = 9/440 T(6, 22) = 1/48 T(7, 22) = 1/49 T(8, 22) = 9/440 T(9, 22) = 7/396 T(10, 22) = 13/660 T(11, 22) = 1/22 T(12, 22) = 7/396 T(13, 22) = 21/1144 T(14, 22) = 1/56 T(15, 22) = 1/66 T(16, 22) = 1/66 T(17, 22) = 3/187 T(18, 22) = 13/792 T(19, 22) = 13/836 T(20, 22) = 7/440 T(21, 22) = 1/66 T(2, 23) = 1/46 T(3, 23) = 1/48 T(4, 23) = 21/1012 T(5, 23) = 1/50 T(6, 23) = 19/966 T(7, 23) = 19/966 T(8, 23) = 17/920 T(9, 23) = 1/54 T(10, 23) = 17/920 T(11, 23) = 1/55 T(12, 23) = 13/828 T(13, 23) = 53/2990 T(14, 23) = 17/966 T(15, 23) = 1/60 T(16, 23) = 1/69 T(17, 23) = 1/69 T(18, 23) = 19/1242 T(19, 23) = 27/1748 T(20, 23) = 7/460 T(21, 23) = 29/1932 T(22, 23) = 15/1012 T(2, 24) = 1/24 T(3, 24) = 1/24 T(4, 24) = 1/24 T(5, 24) = 7/360 T(6, 24) = 1/24 T(7, 24) = 1/56 T(8, 24) = 1/24 T(9, 24) = 1/54 T(10, 24) = 1/60 T(11, 24) = 19/1056 T(12, 24) = 1/24 T(13, 24) = 5/312 T(14, 24) = 1/56 T(15, 24) = 1/60 T(16, 24) = 1/48 T(17, 24) = 1/72 T(18, 24) = 1/72 T(19, 24) = 9/608 T(20, 24) = 1/60 T(21, 24) = 5/336 T(22, 24) = 1/66 T(23, 24) = 1/69 T(2, 25) = 1/50 T(3, 25) = 23/1200 T(4, 25) = 23/1200 T(5, 25) = 1/25 T(6, 25) = 1/54 T(7, 25) = 1/56 T(8, 25) = 1/56 T(9, 25) = 4/225 T(10, 25) = 1/50 T(11, 25) = 19/1100 T(12, 25) = 1/60 T(13, 25) = 14/975 T(14, 25) = 17/1050 T(15, 25) = 1/60 T(16, 25) = 1/64 T(17, 25) = 1/75 T(18, 25) = 1/75 T(19, 25) = 13/950 T(20, 25) = 3/200 T(21, 25) = 1/75 T(22, 25) = 23/1650 T(23, 25) = 1/75 T(24, 25) = 1/75 T(2, 26) = 1/26 T(3, 26) = 1/54 T(4, 26) = 1/52 T(5, 26) = 1/55 T(6, 26) = 11/624 T(7, 26) = 1/56 T(8, 26) = 11/624 T(9, 26) = 19/1170 T(10, 26) = 1/60 T(11, 26) = 9/572 T(12, 26) = 1/60 T(13, 26) = 1/26 T(14, 26) = 4/273 T(15, 26) = 37/2340 T(16, 26) = 1/64 T(17, 26) = 1/68 T(18, 26) = 1/78 T(19, 26) = 1/78 T(20, 26) = 7/520 T(21, 26) = 11/819 T(22, 26) = 1/78 T(23, 26) = 4/299 T(24, 26) = 1/78 T(25, 26) = 17/1300 T(2, 27) = 1/54 T(3, 27) = 1/27 T(4, 27) = 25/1404 T(5, 27) = 23/1350 T(6, 27) = 1/54 T(7, 27) = 22/1323 T(8, 27) = 7/432 T(9, 27) = 1/27 T(10, 27) = 1/60 T(11, 27) = 17/1188 T(12, 27) = 7/432 T(13, 27) = 1/65 T(14, 27) = 5/378 T(15, 27) = 2/135 T(16, 27) = 13/864 T(17, 27) = 1/68 T(18, 27) = 1/54 T(19, 27) = 1/81 T(20, 27) = 1/81 T(21, 27) = 5/378 T(22, 27) = 4/297 T(23, 27) = 1/81 T(24, 27) = 1/72 T(25, 27) = 17/1350 T(26, 27) = 1/78 T(2, 28) = 1/28 T(3, 28) = 13/756 T(4, 28) = 1/28 T(5, 28) = 1/60 T(6, 28) = 1/60 T(7, 28) = 1/28 T(8, 28) = 1/56 T(9, 28) = 1/63 T(10, 28) = 11/700 T(11, 28) = 1/66 T(12, 28) = 5/336 T(13, 28) = 1/65 T(14, 28) = 1/28 T(15, 28) = 17/1260 T(16, 28) = 5/336 T(17, 28) = 1/68 T(18, 28) = 1/72 T(19, 28) = 1/84 T(20, 28) = 1/84 T(21, 28) = 1/84 T(22, 28) = 1/77 T(23, 28) = 33/2576 T(24, 28) = 1/84 T(25, 28) = 1/84 T(26, 28) = 9/728 T(27, 28) = 1/84 T(2, 29) = 1/58 T(3, 29) = 1/60 T(4, 29) = 27/1624 T(5, 29) = 26/1595 T(6, 29) = 25/1566 T(7, 29) = 1/63 T(8, 29) = 1/64 T(9, 29) = 41/2610 T(10, 29) = 21/1450 T(11, 29) = 1/66 T(12, 29) = 19/1392 T(13, 29) = 45/3016 T(14, 29) = 1/70 T(15, 29) = 16/1305 T(16, 29) = 19/1392 T(17, 29) = 27/1972 T(18, 29) = 1/72 T(19, 29) = 1/76 T(20, 29) = 1/87 T(21, 29) = 1/87 T(22, 29) = 15/1276 T(23, 29) = 49/4002 T(24, 29) = 17/1392 T(25, 29) = 17/1450 T(26, 29) = 1/87 T(27, 29) = 37/3132 T(28, 29) = 19/1624 T(2, 30) = 1/30 T(3, 30) = 1/30 T(4, 30) = 1/60 T(5, 30) = 1/30 T(6, 30) = 1/30 T(7, 30) = 13/840 T(8, 30) = 13/840 T(9, 30) = 2/135 T(10, 30) = 1/30 T(11, 30) = 1/66 T(12, 30) = 1/60 T(13, 30) = 11/780 T(14, 30) = 1/70 T(15, 30) = 1/30 T(16, 30) = 1/80 T(17, 30) = 7/510 T(18, 30) = 1/72 T(19, 30) = 1/76 T(20, 30) = 1/60 T(21, 30) = 1/90 T(22, 30) = 1/90 T(23, 30) = 4/345 T(24, 30) = 1/80 T(25, 30) = 1/75 T(26, 30) = 3/260 T(27, 30) = 1/90 T(28, 30) = 1/84 T(29, 30) = 1/87 T(2, 31) = 1/62 T(3, 31) = 29/1860 T(4, 31) = 29/1860 T(5, 31) = 1/65 T(6, 31) = 1/66 T(7, 31) = 25/1736 T(8, 31) = 25/1736 T(9, 31) = 23/1674 T(10, 31) = 1/70 T(11, 31) = 24/1705 T(12, 31) = 1/72 T(13, 31) = 21/1612 T(14, 31) = 3/217 T(15, 31) = 1/75 T(16, 31) = 17/1488 T(17, 31) = 20/1581 T(18, 31) = 11/837 T(19, 31) = 54/4123 T(20, 31) = 1/80 T(21, 31) = 1/93 T(22, 31) = 1/93 T(23, 31) = 1/93 T(24, 31) = 25/2232 T(25, 31) = 26/2325 T(26, 31) = 1/93 T(27, 31) = 37/3348 T(28, 31) = 19/1736 T(29, 31) = 1/93 T(30, 31) = 1/93 T(2, 32) = 1/32 T(3, 32) = 1/66 T(4, 32) = 1/32 T(5, 32) = 7/480 T(6, 32) = 7/480 T(7, 32) = 1/70 T(8, 32) = 1/32 T(9, 32) = 1/72 T(10, 32) = 1/70 T(11, 32) = 23/1760 T(12, 32) = 1/72 T(13, 32) = 5/416 T(14, 32) = 3/224 T(15, 32) = 1/75 T(16, 32) = 1/32 T(17, 32) = 19/1632 T(18, 32) = 11/864 T(19, 32) = 31/2432 T(20, 32) = 1/80 T(21, 32) = 1/84 T(22, 32) = 1/96 T(23, 32) = 1/96 T(24, 32) = 1/96 T(25, 32) = 11/1000 T(26, 32) = 7/624 T(27, 32) = 1/96 T(28, 32) = 5/448 T(29, 32) = 5/464 T(30, 32) = 1/96 T(31, 32) = 21/1984 T(2, 33) = 1/66 T(3, 33) = 1/33 T(4, 33) = 31/2112 T(5, 33) = 1/70 T(6, 33) = 1/66 T(7, 33) = 1/70 T(8, 33) = 1/72 T(9, 33) = 1/72 T(10, 33) = 3/220 T(11, 33) = 1/33 T(12, 33) = 3/220 T(13, 33) = 1/78 T(14, 33) = 23/1848 T(15, 33) = 13/990 T(16, 33) = 1/80 T(17, 33) = 2/187 T(18, 33) = 7/594 T(19, 33) = 47/3762 T(20, 33) = 49/3960 T(21, 33) = 1/84 T(22, 33) = 1/66 T(23, 33) = 1/99 T(24, 33) = 1/99 T(25, 33) = 17/1650 T(26, 33) = 3/286 T(27, 33) = 13/1188 T(28, 33) = 1/99 T(29, 33) = 10/957 T(30, 33) = 7/660 T(31, 33) = 7/682 T(32, 33) = 1/96 T(2, 34) = 1/34 T(3, 34) = 8/561 T(4, 34) = 1/68 T(5, 34) = 31/2210 T(6, 34) = 1/72 T(7, 34) = 29/2142 T(8, 34) = 15/1088 T(9, 34) = 29/2142 T(10, 34) = 13/1020 T(11, 34) = 1/77 T(12, 34) = 13/1020 T(13, 34) = 1/78 T(14, 34) = 11/952 T(15, 34) = 13/1020 T(16, 34) = 1/80 T(17, 34) = 1/34 T(18, 34) = 5/459 T(19, 34) = 23/1938 T(20, 34) = 1/85 T(21, 34) = 1/84 T(22, 34) = 1/88 T(23, 34) = 1/102 T(24, 34) = 1/102 T(25, 34) = 1/102 T(26, 34) = 9/884 T(27, 34) = 8/765 T(28, 34) = 5/476 T(29, 34) = 1/102 T(30, 34) = 7/680 T(31, 34) = 16/1581 T(32, 34) = 11/1088 T(33, 34) = 1/102 T(2, 35) = 1/70 T(3, 35) = 1/72 T(4, 35) = 33/2380 T(5, 35) = 1/35 T(6, 35) = 31/2310 T(7, 35) = 1/35 T(8, 35) = 29/2240 T(9, 35) = 4/315 T(10, 35) = 1/70 T(11, 35) = 1/77 T(12, 35) = 1/84 T(13, 35) = 64/5005 T(14, 35) = 1/70 T(15, 35) = 1/84 T(16, 35) = 1/80 T(17, 35) = 1/85 T(18, 35) = 19/1890 T(19, 35) = 22/1995 T(20, 35) = 1/84 T(21, 35) = 1/84 T(22, 35) = 1/88 T(23, 35) = 1/92 T(24, 35) = 1/105 T(25, 35) = 1/105 T(26, 35) = 1/105 T(27, 35) = 5/504 T(28, 35) = 3/280 T(29, 35) = 41/4060 T(30, 35) = 1/105 T(31, 35) = 43/4340 T(32, 35) = 11/1120 T(33, 35) = 3/308 T(34, 35) = 23/2380 T(2, 36) = 1/36 T(3, 36) = 1/36 T(4, 36) = 1/36 T(5, 36) = 1/75 T(6, 36) = 1/36 T(7, 36) = 1/77 T(8, 36) = 1/72 T(9, 36) = 1/36 T(10, 36) = 1/80 T(11, 36) = 5/396 T(12, 36) = 1/36 T(13, 36) = 29/2340 T(14, 36) = 1/84 T(15, 36) = 1/90 T(16, 36) = 7/576 T(17, 36) = 1/85 T(18, 36) = 1/36 T(19, 36) = 7/684 T(20, 36) = 1/90 T(21, 36) = 1/84 T(22, 36) = 1/88 T(23, 36) = 1/92 T(24, 36) = 1/72 T(25, 36) = 1/108 T(26, 36) = 1/108 T(27, 36) = 1/108 T(28, 36) = 5/504 T(29, 36) = 5/522 T(30, 36) = 1/90 T(31, 36) = 7/744 T(32, 36) = 1/96 T(33, 36) = 1/99 T(34, 36) = 1/102 T(35, 36) = 1/105 T(2, 37) = 1/74 T(3, 37) = 35/2664 T(4, 37) = 35/2664 T(5, 37) = 33/2590 T(6, 37) = 1/78 T(7, 37) = 33/2590 T(8, 37) = 1/80 T(9, 37) = 1/81 T(10, 37) = 1/80 T(11, 37) = 29/2442 T(12, 37) = 1/84 T(13, 37) = 28/2405 T(14, 37) = 1/84 T(15, 37) = 23/2220 T(16, 37) = 27/2368 T(17, 37) = 59/5032 T(18, 37) = 1/90 T(19, 37) = 20/2109 T(20, 37) = 23/2220 T(21, 37) = 103/9324 T(22, 37) = 9/814 T(23, 37) = 1/92 T(24, 37) = 1/96 T(25, 37) = 1/111 T(26, 37) = 1/111 T(27, 37) = 1/111 T(28, 37) = 19/2072 T(29, 37) = 61/6438 T(30, 37) = 16/1665 T(31, 37) = 1/111 T(32, 37) = 11/1184 T(33, 37) = 1/111 T(34, 37) = 1/111 T(35, 37) = 1/111 T(36, 37) = 1/111
Optimal muffin problem solutions occur on an infinite set of linear bounds. For each linear bound there is a continuous range of m/p where the bound is tight, plus a set of discrete optimal instances which converge to one end of the continuous optimal range. The muffin problem should really be considered in projective space r = m/p, and I'll henceforth usually write S(r) instead of S(m,p). Note that S(r) is often continuous in r, while neither of the alternate measures T or U are. We have T(a,b) = S(a,b)/min(a,b) and U(a,b) = abT(a,b). Some choices for unit measures which are continuous in r = a/b are the min, max, or some mean. Because the muffin problem is about the smallest size piece, min(a,b), used by S(a,b) and S(r), seems the appropriate choice of unit measure. Thus "muffins" (smaller parts) have size 1 and "persons" (larger parts) have size r. I noted earlier that S(m,p) <= m/4p for 3p <= m <= 4p. In projective space this result is S(r) <= r/4 for 3/2 <= r <= 2. The bound is tight on (3/2,8/5]. The discrete optimal solutions S(r) = r/4 occur at r = (12+8k)/(7+5k), e.g., S(12/7) = 3/7, S(5/4) = 5/12, S(28/17) = 7/17, ... which converge to S(8/5) = 2/5. I've written a program to calculate S'(r), which is the muffin problem restricted to partitions where all muffins are split into exactly 2 pieces. S'(r) differs from the general solution S(r) only at integral r, where S' = 1/2 and S = 1, and on some subintervals of (1,3/2), where S = 1/3 and S' < 1/3. Here is the beginning of a Fibonacci-like sequence of bounds from my program: S'(r) = (1r-0)/4 = (3/8,2/5] on (3/2,8/5] and at r = (4+8k)/(2+5k) (1r-0)/4 = (1-0r)/2 at r = 2/1 = 2.0000000 S'(2/1) = 1/2 = 0.50000000 S'(r) = (3-1r)/3 = [2/5,1/3) on [9/5,2/1) and at r = (12+9k)/(7+5k) (3-1r)/3 = (1r-0)/4 at r = 12/7 = 1.7142857 S'(12/7) = 3/7 = 0.42857143 S'(r) = (2r-1)/6 = (17/42,7/17] on (12/7,59/34] and at r = (21+59k)/(12+34k) (2r-1)/6 = (3-1r)/3 at r = 7/4 = 1.7500000 S'(7/4) = 5/12 = 0.41666667 S'(r) = (12-5r)/8 = [7/17,13/32) on [148/85,7/4) and at r = (80+148k)/(46+85k) (12-5r)/8 = (2r-1)/6 at r = 40/23 = 1.7391304 S'(40/23) = 19/46 = 0.41304348 S'(r) = (7r-6)/15 = (142/345,265/643] on (40/23,1119/643] and at r = (228+1119k)/(131+643k) (7r-6)/15 = (12-5r)/8 at r = 228/131 = 1.7404580 S'(228/131) = 54/131 = 0.41221374 S'(r) = (73-32r)/42 = [1107/2686,2267/5502) on [9349/5372,228/131) and at r = (1347+9349k)/(774+5372k) (73-32r)/42 = (7r-6)/15 at r = 449/258 = 1.7403101 S'(449/258) = 319/774 = 0.41214470 S'(r) = (61r-60)/112 = (11909/28896,35408/85913] on (449/258,149516/85913] and at r = (10696+149516k)/(6146+85913k) (61r-60)/112 = (73-32r)/42 at r = 764/439 = 1.7403189 S'(764/439) = 2533/6146 = 0.41213798 S'(r) = (1101-487r)/615 = [1555262/3773645,111271/269985) on [1313469/754729,764/439) and at r = (160212+6567345k)/(92059+3773645k) (1101-487r)/615 = (61r-60)/112 at r = 160212/92059 = 1.7403187 S'(160212/92059) = 37941/92059 = 0.41213787 I hope these cases provide some interesting examples for the mixed integer optimization folks to chew on :) I'll give some example optimal partitions from this sequence (40/23, 228/131, 449/258) in a subsequent message.
participants (2)
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Huddleston, Scott -
Michael Reid