. Carnegie Institution of Washington publication. CHAP, ill] PARTITIONS. 131 summed for all solutions of Xi + 2A2 + • • • + p\p = p. At the end of this § 12, he gave other expressions for Cp. He93 later transformed the above formula into p\Cp = \ J. (1 - xn+1). where [XP~\T denotes the coefficient of xp in r, while, after the expansion, 5* is to be replaced by i\. Similarly, for the number Wp of sets of positive integral solutions of a^i + • • • + anxn = p, p\Wp = [>p]{5 - log (1 - xai) •••(!- of")}", which is much simpler to apply than Sylvester's43 formula. He stated (p. 1259) t
. Carnegie Institution of Washington publication. CHAP, ill] PARTITIONS. 131 summed for all solutions of Xi + 2A2 + • • • + p\p = p. At the end of this § 12, he gave other expressions for Cp. He93 later transformed the above formula into p\Cp = \ J. (1 - xn+1). where [XP~\T denotes the coefficient of xp in r, while, after the expansion, 5* is to be replaced by i\. Similarly, for the number Wp of sets of positive integral solutions of a^i + • • • + anxn = p, p\Wp = [>p]{5 - log (1 - xai) •••(!- of")}", which is much simpler to apply than Sylvester's43 formula. He stated (p. 1259) the generalization to two variables: '*» V) = rr^ Op2/9](5 + (5 + log ^)p}9. F. Franklin94 proved that if, in all the partitions of n which do not con- tain more than one element 1, each partition containing 1 be counted as unity and each partition not containing 1 be counted as the number of different elements occurring in it, the sum of the numbers so obtained is the number of partitions of n — 1. Application is made to the distribution of bonds between atoms. A. Cayley95 noted that the partition abc-def of 6 letters into 3's contains 6 duads ab, ac, be, •••, while the partition ab-cd-ef into 2's contains 3 duads. Hence if a partitions into 3's and /3 partitions into 2's contain all 15 duads once and but once, 6a + 3/3 = 15. The solution a = 1, 0 = 3, furnishes an answer of the partition problem: abc-def, ad -be -of, ae-bf-cd, af-bd-ce. Likewise for a = 0, 0 = 5; but not a — 2, 0 — 1. Similarly for 15 or 30 letters. J. J. Sylvester96 considered the e = (w; i, j) partitions of w into j parts 0, 1, • • •, i, the elements of a partition being arranged in non-increasing order, as 3, 2, 2. Without computing e and f = (w — 1; i, j) separately, we obtain e — f = E — F, by counting the E partitions of w in which the initial two parts are equal, and the F partitions of w — 1 in which one element is i. Also, 3=0 F. Franklin97 proved this rul
Size: 1685px × 1483px
Photo credit: © Book Worm / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookauthorcarnegie, bookcentury1900, bookdecade1900, bookyear1902
