OF THE COMPOSITIONS OF NUMBERS. 
863 
Altogether there are I—s — t points at disposal, which may be selected 
as positions for nodes in ways. Suppose + j,^ additional nodes taken, such 
thatji zero-positive and jg positive-zero contacts are introduced into each composition ; 
the number of parts in each composition will be increased byji\ + jV 
We have a one-to-one correspondence between the compositions, having 
« + 1 +ii + is parts, 
t parts without a zero element, 
zero-positive contacts, 
jig positive-zero contacts; 
and those having 
^ + 1 d-ii +>3 parts, 
s parts without a zero element, 
ji zero-positive contacts, 
jg positive-zero contacts. 
We have thus pairs of inverse compositions; the correspondence for the bipartite 
22 is 
= (0,1, 1, 0). 
(1, 0, 1 
, 0), 
(1, 
1, 0, 1), 
(To 12) 
and 
(02 To 
10); 
(11 
OT To); 
(0, I, 0, 1), 
(1, 0, 0 
. 1). 
(1> 
1, 1. 0), 
(n oT) 
and 
(oT or 
20); 
(01 
To TT). 
(0, 1, 1, 1), 
(1, 0. 
1, 1), 
(To n oT) 
and 
(oT oT 
lO To); 
Of the above, two compositions (11 01 10), (01 10 11) may be termed self-inverse, 
29. There are twelve compositions, viz,, those with zero-zero contacts, which do not 
appear in the theory. 
To enumerate these compositions, consider the lines of route with s essential 
nodes or bends j ; every such line must have either s — 1, s, or s -|- 1 bends_| . If 
the allied permutation neither commences with a nor ends with it is s — 1 ; if 
either a commences or ^ ends it is s; if both a commences and 13 ends it is s -p 1. 
The enumeration of the lines of route gives 
I i) (! ~ 1) for s - 1 beods _| ; 
7 ^) (! ID ( * - J) C 7 ® - ■! ■ 
7 ^) ('^ 7 ^ —I • 
