OF THE COMPOSITIONS OF NUMBERS. 
885 
Let one such product with its attached numerical coefficient be 
X 
The number k enumerates a certain number of the permutations which have 6’ major 
contacts, and S k enumerates the entire number. 
The problem is to determine the particular permutations enumerated by the 
number k. 
If and be both greater than zero, the contact must occur in one or more 
of the enumerated permutations. 
If be zero, is absent, while if p>z be zero, a., has no existence, and also X^^ does 
not appear. 
Hence, X^,;^ only appears when both and 2^-2 are greater than zero. It follows 
that the products which involve X21 only enumerate by their coefficients those per- , 
mutations which possess contacts. In fact, the absence of either or a^, by 
causing f® vanish, diminishes the total number of permutations which have a 
definite number of major contacts ; this would not be the case if the terms comprising 
Xji as a factor did not enumerate contacts ; if contacts were not enumerated, 
the vanishing of or would not alter the enumeration of the permutations 
possessing a given number of contacts, for this given number would be complete 
without contacts at all. 
It is next to be shown that a product involving Xof"' comprises exactly .93^, a^ot^ 
contacts. 
Suppose 2h < 521, Vi < ^21- 
Then occurs less than ^gi tunes. 
And, therefore, X2ia2 cannot be raised to the power Soi- 
And, therefore, X^i cannot occur in any term to so high a power as Soi. 
Similarly if pi < Sgi it is evident that X21 cannot occur to so high a power as 
6*21. Therefore, unless both 2^\ p.j are at least as great as X21 cannot occur in 
any term to so high a power as S21. It follows that the products which involve 
X21*'' cannot enumerate permutations possessing fewer than i^i, a2ai contacts. For 
suppose that the permutations enumerated possessed only cr^i, contacts where 
^21 521. If Pi cr P2 be diminished so as to be less than 931 the permutations 
possessing cr^i, contacts would not necessarily be diminished in number, whilst 
those possessing S21 such contacts as well as the products involving X2/'‘ would certainly 
vanish. Hence the products involving Xof"* cannot, through their coefficients, enmrrerate 
permutations possessing fewer than s.,^, contacts. Hence the number k enumerates 
permutations having at least 
