+2}. 1 - x^{i-[t]+3}. ... 1 - x^i
--------------------------------------------------------------,
1 - x. 1 - x^2. 1 - x^3. ... 1 - x^[t]
([t] = [theta])
or of x^n in

1 - x^{2i-2[t]+2}. 1 - x^{2i-2[t]+4}. ... 1 - x^2i
-------------------------------------------------- x[t]^2;
1 - x^2. 1 - x^4. 1 - x^6. ... 1 - x^[t]

hence the expansion

(1 + ax)(1 + ax^3)(1 + ax^5)...(1 + ax^{2i-1})

[t]=i 1 - x^{2i-2[t]+2}. 1 - x^{2i-2[t]+4}. ... 1 + x^2i
= 1 + [Sigma] -------------------------------------------------- x^[t]^2.a^[t].
[t]=1 1 - x^2. 1 - x^4. 1 - x^6. ... 1 - x^2[t]

Extension to three dimensions.

There is no difficulty in extending the graphical method to three
dimensions, and we have then a theory of a special kind of partition of
multipartite numbers. Of such kind is the partition

_________ _________ _________
(a1a2a3...), (b1b2b3...), (c1c2c3..., ...)

of the multipartite number
_______________________________________________________________
(a1 + b1 + c1 + ..., a2 + b2 + c2 + ..., a3 + b3 + c3 + ..., ...)

if a1 >= a2 >= a3 >= ...; b1 >= b2 >= b3 >= ..., ...
a3 >= b3 >= c3 >= ...,

for then the graphs of the parts /a1a2a3..., /b1b2b3..., ... are
superposable, and we have what we may term a _regular_ graph in three
dimensions. Thus the partition (/643, /632, /411) of the multipartite
/(16, 8, 6) leads to the graph

0+------------------------------------ x
|
| ((.)) ((.)) ((.)) ((.)) (.) (.)
|
| ((.)) (.) (.) .
|
| ((.)) (.) .
|
y

and every such graph is readable in six ways, the axis of z being
perpendicular to the plane of the paper.

_Ex. Gr._
___ ___ ___
Plane parallel to xy, direction Ox reads (643,632,411)
______ ______ ______
" " xy, " Oy " (333211,332111,311100)
___ ___ ___ ___ ___ ___
" " yz, " Oy " (333,331,321,211,110,110)
___ ___ ___ ___ ___ ___
" " yz, " Oz " (333,322,321,310,200,200)
______ ______ ______
" " zx,