154 HISTORY OF THE THEORY OF NUMBERS. [CHAP, in 



of nodes is symmetrical in two dimensions) or a^2-symmetry (when the six 

 forms obtained by rotations about the various axes are identical). 



MacMahon, 176 to enumerate the combinations defined by certain laws, 

 would find an operation and a function such that the result of performing 

 the operation on the function gives the number of combinations. Thus, 

 operating with (d/dx) n on x n we get the number n\ of permutations of n 

 distinct letters. Again, let di = d/ddi + did/da 2 + a 2 d/da 3 + , where 

 the a's are the elementary symmetric functions of i, , a n . Using 

 symbolic multiplication as in Taylor's theorem, write D s = dl/sl. Then 

 operating with D,, D nn on (i + + a n ) n we get the number of 

 permutations of a* 1 a* n where STT; = n.. Finally, if we apply D 3 DlDi 

 to the symmetric function (1 4 )(1 3 )(1), where (1 s ) denotes a s = Sai--- a s 

 in partition notation, we get the Sylvester-Ferrers' graph of the partition 

 (3 2 2 1) or its conjugate (4 3 1), according as it is read by rows or columns. 

 The method is successful in solving the problem of the Latin Square 189 in 

 its most general aspect. Cf. Hammond. 217 " 



R. D. von Sterneck, 177 to extend Vahlen's 150 work from modulus 3 to 

 modulus 5, considered the excess \n} h of the number of representations of n 

 by an even number of summands over the number by an odd number of 

 summands, where the summands are distinct and the sum of their absolutely 

 least residues ( 2, 1, 0, 1, 2) modulo 5 has the value h. He proved the 

 recursion formulas 



{k} h = {fc -- 2/i + 3} 3 -*, {k} h = {k - 5h 

 By successive applications of the second, we get 



Hence its value depends on certain [l} j iorj = 0, 1, 2. By Lagrange's 

 theorem, Z{fc}* = or (- 1)' for k =f= or k = (3t 2 t)/2, 



where h ranges over the integers = k (mod 5). This gives a recursion 

 formula for {k} j , j = 0, 1, db 2. Hence we can compute any {k} h . 



M. d'Ocagne 178 found the number of ways s francs can be formed with s 

 French silver coins (5, 2, 1, i, i francs), also when the number of smallest 

 coins is fixed. 



R. D. von Sterneck 179 gave an elementary derivation of the number of 

 decompositions of n into six or fewer equal or distinct positive integral 

 summands, distinguishing 29 types like n = a-\-a-\-(3-\-f3, often with 

 various sub-cases. Thus the results are expressed by many formulas. 



E. Netto 180 employed eight symbols for the various types of combinations 

 and variations, with a prescribed sum, of given numbers taken & at a time, 



176 Trans. Cambridge Phil. Soc., 16, 1898, 262; Phil. Trans. Roy. Soc. London, 194, A, 



1900, 361. 



177 Sitzungsber. Akad. Wiss. Wien (Math.), 109, Ila, 1900, 28^3. 



178 Bull. Soc. Math. France, 28, 1900, 157-168. 



179 Archiv Math. Phys., (3), 3, 1901, 195-216. 

 ""Lehrbuch der Combinatorik, 1901. 



