366 MAJOR P. A. MAGMA HON ON THE THEORY OF PARTITIONS OF NEAIBERS. 
Art. 79. If the partition be into i parts, we can assign 2*“^ different orders 
depending upon the symbols >, and these can all be expressed by conditional 
relations affecting aj, a,, . . . a-, involving the symbol > only. These are not all 
essentially different, as one order does or does not give rise to a different order by 
inversions of parts. Denoting >, < by the letters cl, a, we have for i = 3 the orders 
(Id, da, ad, aa ; the orders del, aa are not essentially different, because interchange 
of a and d combined with inversion converts the one into the other; da, ad are 
essentially different, because this two-fold operation leaves each of these unchanged. 
Hence there are three orders to be considered, and the results have been obtained 
above. 
For ^ = 4 we have the essentially different orders ddd, dda, dad, add. The first 
of these has been obtained ; the other three are solved bv the n functions ; 
o 
1 - rtiX,. 1 _ X.,. 1 - — . 1 - c.,X4 
i>. - a.-,n 
1 
n -;-r- 
- 1 - c,X,. 1 - . I - . 1 - 
■ ■■ a.. 
O 
1 
== ] _ .1 - . 1 - XT . 1 - 
a.. 
a.. 
which reduce to the three expressions : 
1 
1 - Xj. 1 
X ^. 1 - XiX.. 1 - XiXoXoX, 
1 - XiX,,X|X, 
1 - Xi. 1 - XT . 1 - X„X,. 1 - XjXTXo. 1 - XjXoXoX 
1 - X^X^Xs - XiXIX^X, - XiXIXiX, + XiX^X^X, + XfXSXiX 
I - X,. 1 - X,XT. 1 - X,X,. 1 - XjXTX,. 1 - X,X„X,. 1 - XjX.X.X 
and to the three enumerating functions : 
1 .r -f- x~ 
- . 1 - .1 - of .1 - i 
1 -F .r + of + of + V* 
1 - . 1 - .rT 1 - a'T 1 - .r' 
1 -t- -p .f 
1 - rr . 1 
The last real o'eneratinn function that has been written down o-ives the solution 
of the system of conditions 
aoSraj, a.>>aa, a., > ; 
