AND ON COMBINATIONS AND PERMUTATIONS. 473 



4. Since N = rj + (iV — tf), therefore take all such (p — 1) — partitions of A'^ - tj, as have »; 

 for their lower limit ; and with every such set of parts let rj be conjoined as an additional, or p"' 

 part. We shall thus obtain all such p — partitions of A^, as, besides having r; for their lower limit, 

 agree also in containing at least one part equal to ?;. From these partitions, therefore, are excluded 

 all those p - partitions of N whicii have for their lower limit >; + 1. Hence, 



[N, p,]-[N, ;,,^,] = [N-r,, p-t^] (H); 



.-. also IN+ V, P+ I,] - [.V+^, p+ 1,,,] = [A^ joj (ii*). 



5. In [A'^, p^], let N <prj. If negatives be admitted, then let A'^ be the greater negative; 

 and by the addition to »; of a positive quantity greater than r/, and to A^ of a positive quantity 

 greater than prj, let the lower limit and the number to be partitioned be rendered positive. Since 

 no positive integer less than prj can be resolved into p parts liaving a positive integer, rj, for their 

 lower limit, no partition of the kind indicated by the notation can be effected. Therefore, 



when N<py], \_N, p^'\ = (m). 



Then if i; be positive, and p is also a positive integer, [o, yj^] = 0. 



The two following extreme cases, [A'^, 0^], and [o, 0,^] require explanation. 



By (II.) IN+n, 1,] - [A^+ „, J,+,] = [A^ oj. 

 But, when A'' and ») are positive integers, [A'^ + ri, I^] = 1, 



and [A^ + >;, 1, + ,] = 1; 



.-. \_N, oJ = 1 - 1 = (,v). 



Also, by (ii.) [A^, 1^^] - [N, Ij,^.,] = [0, 0^]. 

 But [A'', 1.,] = !; and, by (ill.) [A'', 1^^.,]=0; 



.-. [0, 0;^] = 1 -0= 1 (v). 



Hence also, if A^ = p>j, \_N, pj = [p,;, ;jj = [o, pj] = [/>, ;j,] = I. 



6. Professor De Morgan (as he informs me) has, in the Cambridge Mathematical Journal, 

 traced Equation (ii.) to its consequences, in the case where the number of parts is preserved 

 constant, and the variation is thrown on the number to be partitioned. 



In this ease, [A^, ;j,] - [A^, ;j,^,] = = ^N - t,, p - 1,], 



[N, p,+,] - [A^, p,+,] = lN-r,-i, p- 1,+ ,] = [AT - ,, - P, P - 1 J. 



[N, p,,,] - [A^, ?.,,,„] = [N-rj-e, p- 1,,J = [A^ - ,; - j,e, p - 1,] ; 

 .-. [iV, p,] - [A^, p,,,^,] = S'l [A^ - „ - P^, P - I,]. 



Here >/ is the lower limit of the parts admitted, and r] + 9 + \ is tiie lower limit of tiie parts 



excluded ; that is to say, all those partitions are excluded which have every part greater than t^ + O. 



Write Y' for r) + Q. Then let N = pY + r, r being a remainder less than p, and Y the integer 



A' 

 nearest to, but not exceeding — . When Y becomes Y, [_N, 7^r+i] = ^^ 



.: [N, ;.,] = [A^, py,,] + Sj-^ [N -r,-pz, p- i J 



S^-^lN-n-pz, p-i^-] 



= [A^ y>r+,] + .Vr-''[A^-(1 +plr,-l-\)-pz, p - I] (VI.) 



X 1 



S^--" [N-0 +p[,i- 1]) - P^, P- 1] (v>>.) 



s I 



.-J I' 2 



