MAJOR P. A. MACMAHON ON THE THEORY OP PARTITIONS OP NUMBERS. 379 
a theorem of reciprocity analogous to and including the well-known theorem con¬ 
nected with the partitions of a number on a line. There is also a lattice theory 
connected with unipartite partitions on a line, for the unit-graph of such a partition 
is nothing more than a number of units and zeros placed at the points of a two- 
dimensional lattice, such numbers being subject to the i ^ ^ j descending orders. 
Art. 96. The fact is that the theory of the two-dimensional lattice, the part- 
magnitude being restricted to unity, is co-extensive with the whole theory of 
partitions upon a line. Hence for such partitions we may represent the generating 
function, diagrammatically, in two ways upon a lattice as well as in two ways upon 
a line. 
The two representations upon a line are 
(^ +1) 
(1) 
(1 + 2) 
(2) 
(1 -t 3) 
(3) 
(1 + 4) 
(4) 
(7 -t- m — 
{m — I 
1) 
) 
(1 4- m) _ 
(ni) 
('ll + 1) 
(m + 2) 
{ '/ il -|- o) 
(m 4- 4) 
(/ + rii — 
1) 
(1 4- m.) 
(1) 
(2) 
(3) 
(4) 
(/-]) 
1 
(/) 
Upon a lattice we have 
and at the point marked s we place the factor 
(■^ + 1 ) 
(s) 
The second lattice is obtained by interchange of I and m. 
The product thus obtained is 
s=?+|ii-l j-^'g ^ bs-hs^l — hs_m 
3 c 2 
