CHAP. Hi] PARTITIONS. 129 



* V. Bouniakowsky 83 treated partitions. 



J. W. L. Glaisher 84 considered the number P(a, , q)x of ways of 

 forming x by addition of the elements a, , q, repetitions allowed, and 

 proved that 

 P(l, 3, 5, .)(2z) = 1 + P(l, 2)(x - 1) + P(l, 2, 3, 4) (a; - 2) + 



P(l, 3, 5, -}(2x + 1) = 2 + P(l, 2, 3)(x - 1) 



+ P(l, 2, - - -, 5) (a - 2) + + P(l, 2, . -, 2x - 1)1, 

 P(l, 3, 5, -)x = P(l, 2)(x - 1) + P(l, 2, 3, 4) (a - 1 - 2 - 3) + -. 



Glaisher 85 formed the derivations of a 4 by the rule of L. F. A. Arbogast : 85a 

 a 4 ; a?b; a?c, a*6 2 ; a?d, a?bc, ab 3 ; , 



omitting coefficients. Each term corresponds to a partition of 4. Thus, 

 if a = 1, b = 2, , a*b corresponds to the only partition 1 1 1 2 of 5 

 into 4 parts > 0. In general, from the derivations of a n we see that the 

 number of terms of the xth derivations of a n equals the number of partitions 

 of x into n parts including zero, also equals the number of partitions of 

 x + n into n parts > 0, and finally equals P(l, , ri)x. 



Glaisher 86 gave formulas for checking the tabulation of partitions. 

 The summations extend over all the N partitions of a given number n, 

 while in any partition, x is the number of 1's, y the number of 2's, etc. 



2(1 + x + xy + xyz +)= 22', S(3 - 2xy + Zxyz -.)= r(n), 

 2(1 -2y + 3yz - tyzw + ) = r(n + 1) - T (n), 

 Z{z - 1 - (x - 2)y + (x - 3)yz - -} = N -1, 



where r is the number of different elements in a partition, and r(n) is the 

 number of divisors of n. If Q(a, b, -}n is the number of partitions 

 without repetitions of n into the elements a, b, , and S(l, -, r)n the 

 number of partitions of n into 1, , r in which all but the highest r appears 

 at least once, 



2Q(1, 2, On = 1 + S(l, 2)n + S(l, 2, 3)n + , 

 Q(l, 3, 5, ...)n-Q(l,3,5, )(w-4) 



- Q(l, 3, 5, -)(n - 8) + Q(l, 3, 5, -)(n - 20) + = 1 or 0, 



according as n is a triangular number or not. The excess of the number 

 of partitions of n into an even number of parts over an odd number of 

 parts is ( l) n Q(l, 3, 5, -)n. A partition into a 1's, 3's, 7 5's, etc., 

 is transformable into 7r = a + 3/3 + 57+ -. Express a, 0, in the 

 binary scale : a = 2 a + 2' + , = 2 b + . In the new form of IT 

 no two parts are equal. Hence a partition into odd parts is converted 

 into a partition into distinct parts, and conversely. 



83 Memoirs Imp. Acad. Sc. f St. Petersburg, 18, 1871, 20; 25, 1875 (Suppl.), No. 1 (In Russian). 



84 Phil. Mag., (4), 49, 1875, 307-311. 



86 Report British Assoc. for 1874 (1875), Sect., 11-15; Comptes Rendus Paris, 80, 1875, 255-8. 

 850 Calcul des derivations, Strasburg, 1800. See papers 46, 102, 198. 

 88 Proc. Roy. Soc. London, 24, 1875-6, 250-9. 

 10 



