644 
MAJOR P. A. MACMAHON ON THE 
Art. 34. The number of lines of route which possess exactly s 31 ya bends and no 
/3a bends is 
Also, by putting X 3l = X 33 = X in the generating function, we find that the number 
of lines of route which have s y/3 and ya bends but no /3a bend is given by the 
coefficient of a. p fi q y r in 
_ (ay + &y — gffiyV _ 
(1 - ay + 1 (1 - f3) s + 1 (1 - y) s + 1 ’ 
Art. 35. It will be convenient to give the complete correspondence in the case of 
some simple tripartite number, say, 222. 
n O 
^ 31 3: 
3 Permutation. 
Composition. 
Partition. 
Number partitioned. 
o o 
o 02 2 
“PT 
(222) 
(22 s ) 
44 
1 0 
afiryay 
(121 101) 
(12 22) 
34 
1 0 
a/3 2 y 2 a 
(T22 100) 
(12*) 
24 
1 0 
ft ~ya~y 
(02l 20T) 
(02 22) 
24 
1 0 
a/3ya/3y 
(111 111) 
(TT 22) 
33 
1 0 
aya/3 2 y 
(101 121) 
(TO 22) 
32 
1 0 
Pya'Py 
(Oil 211) 
(01 22) 
23 
1 0 
PYa* 
(022 200) 
(02 2 ) 
04 
1 0 
aPy z a.p 
(112 110) 
(Iffi) 
22 
1 ' 0 
ya 9 -p 2 y 
(001 221) 
(00 22) 
22 
1 0 
ay~aP' 2 
(102 120) 
m 
20 
1 0 
Py°a-P 
(012 210) 
(0T 5 ) 
02 
1 0 
y 2 a~p 2 
(002 220) 
(bo 2 ) 
00 
0 1 
a i Pypy 
(211 Oil) 
(2T 22) 
43 
0 1 
a~Py~p 
( 2 T 2 "bio) 
(2T 2 ) 
42 
0 1 
a~yP~y 
(201 021) 
(20 22) 
42 
0 1 
a yp~ 
(202 020) 
(20 s ) 
40 
1 1 
apyayp 
(111 101 010) 
(TI 2T) 
32 
1 1 
aPyPyrx. 
(ITT oTT Too) 
(U 12) 
23 
