154 MAJOR P. A. MAOMAHON: MEMOIR ON THE 



3. The line of route above depicted denotes a "principal composition" of the 

 bipartite number (75), viz., 



(21, 33, TT, 10), 



and, in general, some principal composition of the bipartite number (mn). 



4. It also denotes the permutation 



a'/Sa'jS'ajSa 



of the letters in the product a 7 ^ 5 ; and, in general, some permutation of the letters in 

 the product M y3 n . 



5. The line of route divides the lattice into two portions, each of which denotes the 

 Sylvester-Ferrers graph of a partition of a unipartite number. 



Consider, for example, the portion of the lattice bounded by ADEFGHICB. We 

 obtain the graph of a partition in two ways : 



(i) By placing a unit (or node) in the centre of each square contained in the 



bounded area ; thus 



1 



11 



11 



11 



11111 



denotes the partition 54111, or its conjugate 52221 of the number 12. 



In general, we thus obtain a partition of some numbers into n, or fewer, parts, the 

 part magnitude being limited not to exceed m, and its conjugate a partition of some 

 numbers into m, or fewer, parts, the part magnitude being limited not to exceed n. 



Similarly the remaining portion of the lattice denotes some partition and its 

 conjugate. 



(ii) By placing a unit (or node) at the centre of each segment to the right hand of 

 the points A, E, J, K, G, I respectively ; thus 



1 



11 

 11 

 11 



11111 

 1111111 



denotes the partition 6522211, or its conjugate 752221 of the number 19. 



In general, we thus obtain a partition of some numbers into m parts, the part 

 magnitude being limited not to exceed n+1, and its conjugate, a partition having m 

 for the highest part, the number of parts being limited not to exceed n+1. 



The remaining portion of the lattice may be similarly interpreted. 





