362 MAJOR P. A. MxcMAHON ON COMBINATORIAL ANALYSIS. 



since (d/dx)* a? = n ! ; but once we observe the way in which d/dx operates upon x" 

 we require no previous knowledge of the result to aid us in the design. Conceive x" 

 written as a product 



XX 



xxxxnn . . . 



the operation of d/dx consists in substituting unity for x in all possible ways, and 

 summing the results obtained. 



- x* = 1 . xxx ... + xlxx . . . + xxlxx ... + . . . = nx"" 1 . 

 dx 



We have, in fact, to perform n operations of substitution ; let us select one of these, 



say 



xxlxx . . . 



and denote the minor operation, by which it has been obtained, by the scheme 



SL 



the suffix a denoting that the first operation of d/dx has resulted in the appearance 

 of the unit. 



To obtain (xxlxx . . .) we have n 1 minor operations by which x is replaced 



CL3C 



by unity in all possible ways. If one term obtained be 



X 



Ixlxn . . . 

 the operations by which this has been reached may be denoted by the scheme 



and by proceeding in this manner we finally reach a lattice, square and of n z compart- 

 ments, which is the diagrammatic representation of one of the n I combinations of 

 minor operations which results from the operation of (d/dx)" upon x*. If we transfer 

 the lj, 1 . . . to the top row we see that to each diagram corresponds a permutation 

 of the n different letters a, 6, c, ... Moreover suppressing the letters a, 6, c, ... 

 we see that we have solved the following problem, viz. : To place n units in the 

 compartments of the square of order n, so that each row and each column contains 

 one and only one unit. In general we find that the problems that can be solved have 

 some simple definition upon a lattice, as in the present instance. Writing a, b, c, . . . 

 as i, a 2 , a s , . . . the suffix of the letter is given by the row and the place in the 

 permutation by the column, so that to a s standing t tit in a permutation would correspond 

 a unit in the s th row and t ih column of the lattice. 



It may be remarked, and will afterwards appear, that in general many different 

 designs of operation and function are appropriate to a particular problem. 



