CHAP, ill] PARTITIONS. 149 



t , _g_ . g 6 I g 12 | 



m " 23 



1 - g 5 "- 2 



71=1 



where, on the left, the exponents in the numerators are n 2 and n(n + 1). 



G. Brunei 157 considered two sets of n points such that from each point 

 of each set issue two bonds connecting it with two points or a single point 

 of the other set. Each such configuration can be considered as the result 

 of the juxtaposition of polygons of 2ki, -, 2k r sides, where 



ki + + kr = n. 



Regard two configurations as identical if, after a permutation of the points 

 of each set, the bonds are in the same order in the two. For the number 

 h n , r of configurations relative to n and r, 



fin, r ~ "'Ti1, r 1 "T" "n r, r> 



J. Hermes 158 noted that the number of compositions fas by Mac- 

 Mahon 154 ] of m into k parts ^ p is ( M+ *l!r* p ). There are 2 m ~ l compositions 

 of m; each defines the elements of a Gauss Klammer [a, -, p], occurring 

 in continued fractions (Gauss 24 of Ch. II) ; they give the 2 m ~ 2 Farey numbers 

 of the (m l)th set, each taken twice [see Vol. 1, p. 158 of this History]]. 



Hermes 159 generalized Euler's 9 formulas on the number of partitions. 

 If s, t, n are integers ^ 0, let E 9 , t (n) = E t , s (n) be an integer such that 

 E(G) = 1, E m (ri) = if n > 0, and 



E s , t (n) = E s , t (n - f) + E s , ^(vi). 



For t = 0, E St (n) is the number of partitions of n + s into s positive parts. 

 Several recursion formulas are proved, including 



E s , t(n) = Z E s - h , t (n- s + K), 



d-l 



.# a _i, Ce - ks} = E s , t (x) E a , t (x ds). 



k=0 



The number of partitions of n + x 1 into x 1 terms chosen from 

 1, , s + 1 is 



5 / / /? - 



A ( \ V ( t^hTT 1 I & I i ' " 



4=0 V \ 2 



unless n > sx s, when the sum is zero. Properties of the ^.'s are given. 

 A. Thorin 160 asked for the integer k for which the number of sets of 

 positive integral solutions of a&i ++ CL n x n = b, x : + + x n = k 

 is a maximum. 



157 Proces-verbaux des seances soc. des sc. phys. nat. de Bordeaux, 1894-5, 24-7. 



158 Math. Annalen, 45, 1894, 370-80. 



159 Ibid., 47, 1896, 281-297. 



160 L'intermediaire des math., 1, 1894, 181-2. 



JJ = A*_i, 



