THEOEY OF THE PARTITIONS OF NUMBERS. 161 



In the former case the partitions of highest part m and n parts (zero not excluded) 

 are enumerated by 



mn + 2 /m + n 

 m+l \ m 



In the latter the compositions of the number w, having n parts (zero excluded 

 because ft n = +!), are enumerated by 



w 2n+2 /wl \ 

 w-n+1 \n-IJ' 



Ex. gr. for w = 6, n = 3, we have the five compositions 



411 



321 



312 



231 



222 

 which satisfy the given inequalities. 



In this Section I have shown the connexion between a well-known question in 

 probabilities and varioiis other combinatorial questions in preparation for the 

 generalization to which I now proceed in Section 2. 



SECTION 2. 



15. In the Second Memoir on the Partitions of Numbers I broached the subject of 

 the two-dimensional partitions of numbers. I start with any Sylvester-Ferrers graph 

 of an ordinary one-dimensional partition say 



and I consider the parts of the partition to be placed at the nodes in suchwise that 

 the numbers in the rows, read from West to East, and also in the columns read from 

 North to South, are in descending order of magnitude. Thus 



433222 



3222 



2111 



21 



2 



is a two-dimensional partition of the number 35. 

 YOL, ccix, A. T 



