MAJOE P. A. MAcMAHON: MEMOIR ON THE 



The Memoir referred to contained some striking results in the theory, but the 

 general result as conjectured and verified in numerous instances remained unproved. 



The present paper is mainly concerned with the partitions into different parts 

 placed at the nodes of any graph, and with the associated question in probabilities, a 

 generalization of that of Section 1. 



Taking any graph of n nodes and any n different integers, the inquiry is as to the 

 number of ways of placing the numbers at the nodes so that the descending orders in 

 the rows and columns, as above defined, are in evidence. 



Consider in detail a simple case that of six different numbers at the nodes of 

 the graph 



We find the 16 arrangements 



the second row of eight arrangements being the conjugates of those in the first row 

 because the graph is self- conjugate. 



16. The problem is immediately transformable into one concerned with the 

 conditioned permutations of the six numbers in a line. 



Take the form 



654^ 



32 * 



;t 



and suppose the six numbers to be written down in a line so that four descending 



orders 



654^ 



32 -* 

 631* 



52 * 



are in evidence ; I say that there is a one-to-one correspondence between such 

 permutations and the two-dimensional partitions under investigation. 

 To see how this is, take any one of the 16 arrangements 



632 



51 



4 



