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Mr. WARBURTON, ON THE PARTITION OF NUMBERS, 



number of the kinds from which the ^< Elements are taken, shall he z ; or this additional limitation, 

 that „j of the x kinds shall each contain v elements ; 



m' v' ; 



m" V ; 



and so on ; and the Elements from which such Combinations are to be formed, may admit either of 

 limited, or unlimited repetition. 



10. If the given Elements are of s kinds, and may be repeated in each kind without limit, the 

 Coefficient of x", in the product of the geometrically progressing Polynomes, will consist of terms 

 in which there are u elements of one kind, ... of 2 kinds, ... of » kinds, ... z never exceeding u, 

 and finally, when ti becomes equal to, or greater than s, becoming equal to s. Consequently, the 

 models or types, after which these several terms in the Coefficient of x" are formed, will depend 

 altogether on the partitions of the number u into 1, 2, 3, ... z parts. If u < s, the number of 

 these terms will depend on the number of the partitions of u enumerated in the expression, 

 [m, li] + [ti, 2,] + [m, Ml] = [-2 M, M,]. 



When u becomes equal to s, the number of these partitions will be [2 s, s]. When u > s, the 

 number of the partitions will be 



[m, 1,] + [m +2i] + [?«, «,] = [u + s, s,]. 



See Article 7, Section I., of the present Paper. 



Thus, if the Elements are of 6 kinds, and they are to be combined together 7 at a time, there 

 will be in all [ 13, 6] =14 types, in accordance with which all the Combinations, containing 

 7 Elements each, will have to be constructed ; and these types are the following partitions of the 

 number 7. 



10.* Let one of tiie z- partitions of u be 



V, V, V, (to) v', «', v', (m) v", v", v", 



so that m + m' + m" + = z, 



and w t! +m' v + m" v' + = «. 



(m"), &c.. 



