884 
MAJOR P. A. MACMAHOX OR THE THEORY 
M'hich possess exactly s major contacts, is given by the coeflficient of 
. . . «/“ 
in tlie product 
-f X (cto + . . . + ^ («3 + • • • + • • • 
{«! + + • • ■ “ 1 " 
I am not aware if this problem has been previously solved in this form, or, mdeed, 
if it has ever been attacked before.* 
65. “ The number of permutations of the letters in the product 
•^1 > 
which possess exactly s major contacts, is equal to the number of permutations for 
which 
^3 + + • • • + = 5, 
Vt denoting the number of times that the latter occurs in the first 
+ i^3 + • • • + ‘Pt-\ 
places of the permutation.” 
66 . It is easy to obtain a refinement of these theorems which has been fore¬ 
shadowed by the detailed bipartite and tripartite cases which have preceded. Major 
contacts in a joermutation, and, consequently, also the essential nodes along a line of 
route, are of j different kinds. 
Consider the product 
(“l + + Xg^ag -f- . . . -fi (a;^ -f + Xg^ag + + . . . + X,,2a„)^- 
. . . (cij “h o£o “h . . . ~{~ X„ * {,^1 T" d” • • • “h • 
The coefficient of . . . a/" consists of a number of monomial products of the 
quantities X, each attached to a numerical coefficient. Of these products a certain 
number are of a definite degree s. The sum of the coefficients of these products 
gives the number of permutations of the letters in 
afi'a/- . . . a/" 
which possess exactly s major contacts. 
* See a paper by H. Foeiy, M.A., “On Contact and Isolation, a Problem in Permutations,” 
‘ Proceedings London Mathematical Society,’ vol. 15. 
