THEORY OF THE PARTITIONS OF NUMBEBS. 81 



Let the Greek letters a, , y be associated with the first, second, and third rows, 

 respectively, and" consider each number in the lattice in succession in descending order 

 of magnitude. Thus, beginning with 6 : since it is in the first row I commence a 

 succession of Greek letters with a. ; passing to 5, since it is in the second row, I follow 

 with )8 ; then 4 gives y, since it is in the third row ; then 3 gives a ; 2, /8 ; and finally 

 1 gives a. 



In this way I obtain a permutation of the letters in a 3 /3 a y where the exponents 

 3, 2, 1 enumerate the nodes in the successive rows of the lattice. This permutation 

 possesses the property : 



" If a dividing line be made between any two adjacent letters of the permutation, 

 the succession of letters to the left of the dividing line is like the whole 

 permutation, such that a occurs at least as often as /?, ft at least as often 

 as y ; in other words, the numbers which specify the occurrences of a, ft, y, 

 are in descending order of magnitude." 



In fact, if the process of forming the Greek letter succession (or permutation) be 

 arrested at any point, the lattice numbers that have been dealt with occupy a set of 

 nodes which also constitute a lattice, complete or incomplete. 



It follows, of course, that the first letter of the permutation must be a. The lattice 

 arrangement of numbers is recoverable from the permutation, for it is merely 

 necessary to write the numbers in descending order underneath the letters when we 

 see that the successive lattice rows are indicated by the letters a, y8, y, respectively, 



a)8ya)8a 

 654321' 



The process is thus unique, and there will be as many different Greek letter 

 permutations having the properties above specified as of arrangements of unequal 

 numbers at the nodes of the lattice having the specified descending orders. 



Every Greek letter permutation can be separated into groups, each of which contains 

 letters in alphabetical order; in the case before us this is accomplished by two 

 dividing lines 



each of which separates a letter from one which follows it, but is prior to it in 

 alphabetical order. 



I associate a power of x with each permutation by taking for the exponent a sum 

 of numbers pi+p*+p 3 +..., where p. denotes that the a tb dividing line has p, letters 

 to tKe left of it. Thus in the above instance p l = 3, p, = 5, and the associated power 

 of x is x 3+6 = a; 8 . 



VOL. CCXI. A. M 



