i2g 



VI On the k-partitions of N. 



By the Rev. Thomas P. Kirkman, A.M., Rector of Croft- 

 * with- South worth. 



\_Read January 10th, 1804.] 



Any system of k numbers written in the order of their 

 magnitude, whose sum is N, and of which none is less than 

 flf, may be represented thus ; 



{a-\-x{)+(a+Xi+X2)+{a'\-Xi-\-X2'{-X3)+ 



•¥(a-\-Xi-\-X2+ . . . +a:A)=N. 

 where X1X2 &c. may be anything positive or zero ; for none of 

 these k numbers is less than the preceding one, but exceeds 

 it by any difference which is either positive or nothing. And 

 it is plain that no element x^ can be altered in value, the 

 equation remaining true, without giving rise to a new system 

 of k numbers whose sum is N, none of them being less than a. 

 From this equation, written under the form, 



A-i+ii+A— l-a;2+A^-ar3+ +3a:ft.2+2a;^i+rrfc=N, (A) 



many interesting properties can be deduced. Its solutions, 

 in null or positive values of x, x^ &c., are in number equal 

 to that of the different ways in which N can be broken into 

 k whole numbers none less than a. I shall confine myself 

 here to an important deduction from A. Writing it thus, 



k—i-a+Xi-^-k — 2'X3+k—'S'Xi-\' +2'x^^i+Xk 



= N_a_to; (B) 



we see that the number of solutions is the same in B as in A; 

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