76 . MAJOR P. A. MACMAHON: MEMOIR ON THE 



representation by nodes upon a three-dimensional lattice, just as for partitions on a 

 line there is a graphical representation by nodes upon a two-dimensional lattice. 

 is convenient to replace these nodes by units, and to regard partitions on a line as 

 being in one-to-one correspondence with partitions in a plane when the part 

 magnitude of such is restricted to be not greater than unity ; thus, instead of saying 

 with FERRERS that 



is a graphical representation of the line partition 321, I regard the plane partition 



of units 



111 



11 



1 



as being in one-to-one correspondence with the line partition. 



Just so the plane partition 



331 



22 



1 



is graphically represented by piles of nodes perpendicular to the plane of the 

 paper, say 



(0) (0) 

 0. 



or we may replace the nodes by units, and say that it is in one-to-one correspondence 

 with a space partition, the part magnitude being restricted to unity. The plane 

 partition arises by projection of the space partition upon one of the co-ordinate 

 planes, just as the line partition arises by projection of the plane partition, with which 

 it is in correspondence, upon one of the co-ordinate axes. 



Every two-dimensional graph of nodes may be interpreted either by rows or by 

 columns, and every plane partition of units may be projected in two ways. The 

 graphs in solido admit of one, ttvo, three, or six readings. 



In previous papers I omitted to notice that a three-dimensional graph may admit 



of two readings. The omission came to my notice when I was trying to verify that 



the number of partitions of w in piano the numbers of rows and columns, and also the 



t magnitude, being unrestricted is given by the coefficient of a-" in the ascending 



expansion of the algebraic fraction 



1 

 11 -a-) (1-0-7 (1 -*T (I-* 4 ) 4 ... ad inf. ' 



