THE FOUNDATIONS OF A NEW THEORY. 365 



This, the simplest example that could be taken, shows clearly the great value of 

 the operator D w as an instrument in combinatorial analysis. 



Example 2. It follow from the first example that if STT = n, 



Example 3. Consider 



D 4 (2) 2 (l) 2 - 



We are concerned with the partitions of the number 4 into 2,2 and 2,1,1, and 



If now we operate with D 3 we have to take account of the partitions 1,1 and 2 of 

 the number 2, and we find 



D 4 D 2 (2) 2 (I) 2 = 

 and we have the result 



(2)*(1) 2 = - +3(42)+ 

 as a consequence. 



Similarly, reversing the order of the operations 



and D 4 (2)(l)(l) = D 4 (2)(2) = 1, 



verifying the previous result. 



If no partition of TT can be picked out in this way from the partitions of the func- 

 tions forming the product, the result of the operation is zero. 



Example 4. 



D 2 (l')> = D a (l)(l*) = (!)(!) = (I) 3 . 



It is important to notice here that a unit is erased from (I 2 ) = (11) in only 

 one way, and that for present purposes a number of similar figures enclosed in a 

 bracket are to be considered as the same, and not different ; we have already seen 

 that when the figures are similar, but in different brackets, they are, for the purpose 

 of selection, to be considered as different figures. 



Observe that since 



D 2 (l*) =(!)* = (2) + 2(1*), 



we may say that 



(17 = (2*) + 2(21*) + ---- 



the terms to be added on the right for the full expression being such as do not contain 

 a figure 2. 



To obtain the lattice representations, suppose IT,, 7r 8 , ir 3 , . . . ir, to be in descending 

 order, and thus to be an ordered partition of the number n. 



