518 
MAJOR MACMAHOR OR SYMMETRIC FUNCTIONS 
In particular 
^10 ~ Gr(jo), 
^01 ~ Gr(oi), 
^20 ~ ^(To 2) H“ Gr(20)) 
Clil — Gr(!ooI) “h (^(n)j 
^02 — ^1(012) + G'(02)- 
63 . The relations between the partition g operators and the partition G operators 
are of great interest. 
Recalling the equivalence (Arts. 53 and 41 ) 
^ (-f-^(S 7 r-T)! 
TTi ; TTo ! . 
^ . . .) 
= ^ 
Stt — 1 
TTi 1 TT, 
g" 
I ^ Pih ^ Pili 
which should be compared with the algebraical result 
(-)--^( 27 r-l) ! 
Trj! TT^! . . . 
^ (Vtt - 1)! 
TT^! TTg! 
there arise the relations 
S'do) — Gtio — G(io), 
9{Wi) — Gtqi = G(oi), 
5^(102) ~ 2^(20) — GtIO^ 2Goo ~ Gr(xo)^ 2 G(Io2) 2G(2o), 
P'(IooI) g(n) — ^10^01 ~ ^11 ~ Gr(io)G(oi) G(iool) G(n)5 
P'(to2) — 3 p'( 2 oio) + 3 g(so) — Giq^ — 3G20G10 + 3 G; 
30 J 
= G 
(10) 
3G(ib2)G(io) + 3 G(io 3) — 3 {G(^)G(io) — G(2oio)} + 3G| 
(30) j 
and so forth. 
64 . Now consider the relation last written. 
I say that it may be broken up into three relations, viz.;— 
9 (m — Gr(To)^ — 3G(io2)G(io) + 3G(ibs), 
9(WTu) — CT(20)G(ib) G(2 oIo)3 
9 {S 0 ) — Gr( 30 ) ; 
for suppose an operand to be composed of separations of a separable partition 
(10'"“ 20’"“ 30 '""“. . .), the performance of the operations on the two sides of the relation 
