•174 mk. warburton, on the partition of numbers. 



Thus, [31, 5j = .S".' [20 - 5z, 4,], 



z 



= [20, 4,] + [15, 4,] + [10, 4j] + [5, 4,] ; 



i.e. 101 = 64 + 27 + 9 + 1. 



7. By a different transformation of the equation of differences, (11), we arrive at a different 

 summation ; in which the number remains constant, while the parts vary. In that equation, if we 

 write q for p, we have, 



[iV + »;, 9 + 1] - [N, 9,] = [iv + v, 9 + 1] = [N-q>h 9+1]; 



1 1+1 ' 



[iV + 2 V, 9 + 2] - [A'' + »7, 9 + 1 ] = [JV + 2 ,;, 9 + 2] = [iV - 9 >,, 9+2]. 

 >i 1 1+1 ' 



[N + pri, q +p'] -lN+(p - 1)11, 9+ (p - 1)] = \_N+prj, 9 + p] = [N - q>i, 9 + p] ; 



.-. [N + p,,, q+p] - [N, 9j =S^\_N-q,,, q+z]. 



n ^ 1 



Now [N, 9 ] vanishes, either when A^ < qtj, or when q = 0; the exception to the latter case 

 being \rhen N = qtj. \i N < qt); then, since 



\_N + pv,, 9 + p] = [iV - 9 V + 9 + ;J> 9 + ?J] = 0> 



and [iV, ^rj = [iV-9^ + 9, 9,] =0, 



it follows that 6*^' \_N - 9»;, 9 + «] =0. 



^ 1 



And we have only + = 0. 



But if 9 = 0, 



[AT + p, p,] - [A^, 0,] = ^f [A^, z,-\ ; 



z 



.: IN + p, p,] =.S':[A^, s;,] (vm), 



z 



or [A^, p.] =^;[Ar-p, .-,] (vui*); 



.-. also [N, P,]^S^lN-p,,, z,-] (IX). 



z 



Thus [31, 53] = [16, 0] + [16, 1] + [16, 2] + [16, 3] + [16, 4] + [16, 5]. 

 Or, 101 = + 1 + 8 + 21 + 34 + 37. 



The following very elementary proof of this proposition has also suggested itself to me. 

 We shall exhaust all the ways of resolving A'' into p parts, having 1 for their lower limit, 

 if we take 



1st, p - 1 units, and the remainder N - (p - 1) entire, not less than 2. 



2d, p - 2 units, and the Bi-partitions of the remainder N - p + 2, not less than 2. 



7«thly, p - m units, and the m - partitions o{ N - p + m, not less than 2. 



Lastly, p - p = units, and the p - partitions of A'' - p + p = A'', not less than 2*. 



■ When ?)> — , the greatest value of m is N—p; and the 



partition of N, corresponding to that value, is [(2^ - JV) units, 

 and ( A^-p)tepetitions of the number 2]. As regards \\\e number 



of the partitions, the two 

 comprehended in formula (viii*) 



N N 



cases of p not >-^. and P> ^) are both 



