476 ■ Mr. WARBURTON, ON THE PARTITION OF NUMBERS, 



on the (n + l)"" line of the same diagonal may be found; and so in succession, to any required 

 extent, until they become constant : (vide § 9). 



The consequence of the preceding equation is, that any term in the table, say that on the p"" 

 line, in column N, is equal to the sum of all the terms from line to line p inclusive in column 

 N — p ; which is the column at the same distance backwards from column N, that the line o is 

 from the line p. 



y. If in the table we draw a zig-zag line from [o, O] to [12, 6], it will be seen that all the terms 

 below that line are of constant recurrence, and are identical with the numbers 1, 1, 2, 3, 5, 7, 11, 



15, &c., which arise from the summation respectively of all the terms in the columns 0, 1, 2, 



3, i, 5, 6, 7, &c. 1 proceed to explain this. Let any diagonal line proceed from the head, say 

 of column J, advancing simultaneously one column and one line. AVhen that diagonal cuts the line 

 p, N will be equal to A + p. 



Now [A + p, p'] = S^\_A, Z-], 



and whenp = A, or > A, its value becomes constant, and is [_2A, A] = [A, o] + [A, l] + ... [A, A], 

 that is, it becomes equal to the sum of all the terms in column A. Thus one-half of the whole 

 table is occupied by terms = ; and an additional fourth of it by these constants ; and were it 

 thouo-ht requisite to compute a table of the partitions of numbers, it is only the terms that occupy 

 the reraainino- fourth of the whole space of the table, that would actually require computation by 

 the method of differences : and of this fourth the three first lines are so obvious, as merely to 

 require being transcribed. 



SECTION II. 

 0?i Comhinations. 



1. The well-known Theorem in Combinations enables us to determine in how many different 

 ways u elements can be taken at a time out of s elements, all dissimilar. It is the coefficient 

 of .r" in the developed power of the binome, [l -i- a?]', which, in this case, affords the solution of 

 the problem. 



2. In the first case of combinations which I now propose to investigate, the combining elements 

 are also of s different kinds ; but there may be more than one element of the same kind : for instance, 

 a of the elements A, (3 of the elements B, and so on ; and the question proposed is, — In how many 

 different ways u of the said \_a + j3 + &c.] = cr elements can be taken at a time, on condition that 

 those which are plural in their respective kinds, may be repeated in the same combination ? 



3. Combining elements of the form proposed are found in the s geometrically progressing 

 polynomes, 



[l + Ax + A'x- + ^"j;"] X [l X Bsc + fi'**^] X &c., 



and all the possible combinations of these elements, taken 0, 1, 2, , m, cr, at a time, 



are respectively found aggregated, each with a positive sign, in the coefficients we obtain of 



y, a?', .x^ x", , x", 



when the product of the said polynomes is developed according to the powers of x. That develope- 

 ment, supposing all the coefficients to be complete, is of the form 



\ + S\_A']x + S{A'' + AB']x'+ SIA'^ A'B + ABC^x^+ 



+ S lA" + A"-' B + A"-- (.B-' + BC) + &c.] .r" -i- 



