880 
MAJOR P. A. MACMAHON ON THE THEORY 
56, Order 3. 
" { 1 - 
— 1 
Si ( 2«1 + «2 + ^ 3 )} { 1—^2 ( 2^1 + 2 a 2 + « 3 )} {1 - % i 2 «i + 2«2 + Sog )} 
_ 1 _ 
1 2 S^C^o ”i~ ^'3^3 Sg'S^^g^l ““ ~1” '5o*^g ^1 ) 
§1 («2 + gtg) §2 f2«^ + ag) Sg (2^| + 2a2) 
X 
1 + 
+ 
+ 
+ 
+ 
+ 
1 — -Sj (2aj + «2 + ag) 1 — §2 (2^2 + 2«2 + «3) 1 — Sg (2«1 + 2^2 + ' 2 x .^ 
_ S ^ S .^ Ci ^ (2«^ + «2 + ^3)_ 
{1 - (2«1 + «2 + «3)} {1 - ^2 (2«1 + 2«2 + «3)} 
252Sga2 (“1 + ^3 + ^3) 
{1 — Sj (2«^ + ^2 + « 3 )] {1 — Sg (2«^ + 25<2 + 2 «3)} 
_ 2-g2S3«i ( 2 aj + 2«2 + ^ 3 ) _ 
{1 — §2 (2«j + 2a2 + *3)} {1 — Sg (2«j + 2«2 + 2*3)}_ 
57. The general algebraical result, interpreted arithmetically, shows that 
JL___ _ ____ 
^ {1 — S;^ (2a^ + ci.2 + ... + a«)} (1 — ^2 (2“^! + 2«3 + . . . + a„)} • • • {1 — S„ {2u^ + 2«2 + . . . + 2«„)} 
< 
is a generating function for the enumeration of the compositions of multipartite 
numbers. 
58. The original generating function of tlie earlier sections has by the addition of 
ineffective terms become factorized, and can thus be dissected for detailed 
examination. 
The process seems to be analogous to the chemical operation bj' which the addition 
of a flux causes an element to be the more easily melted. 
As a direct consequence of the geometrical representation of the compositions of 
multipartite numbers on a reticulation it is of great interest. 
59. To resume. The number of compositions of the multipartite number PiP^ ■ • • Pn 
is the coefficient of 
a.p'- . . . a,/" 
in the expanded product 
1 (2a^ , -j- (2a^ + 2ao + . . . + a,)P-^ . . . {2a.^ + 2a^ + . . . + 2a.,)P\ 
or we may say that it is the coefficient of the symmetric function 
S ct.p’'- . . . a,/'* 
in the development of the symmetric function 
2 ^ (2ai + “2 + • • • + (2ai + 2a2 + . . . + + 2a2 + . . . + '2a^P' 
in a sequence of monomial symmetric functions. 
