20
GENERALIZATIONS OF THE PERRON-FROBENIUS THEOREM
Thus Corollary 3.3 shows that S is not array-admissible for n.
Corollary 3.11 motivates the following definition.
DEFINITION 3.12. A set S c {1,2,... , n} satisfies condition G for the integer
n if there does not exist a subset {pi,P2,P3,P4,P5}, where 1 pi n for 1 i 5,
Pi 7^ pj for 1 i j 5 and
1. Pi + P5 n for i ^ 5 and pi + p2 + £3 2(n - #4).
2. gcd(pi,p5) = 2 for 1 i 4, gcd(p;,p4) = 2 for 1 i 3 and gcd(pi,pj)|4
for 1 i j 3.
An example of a set S C {1,2,... , 32} that does satisfy the conditions gener-
alized C, D, E and F but does not satisfy condition G (and hence is not an array
admissible set for n = 36) is given by {12,14,16,20,34}.
To summarize the results so far, we have the following result:
THEOREM
3.13. // a set S C {1,2,... ,n} is array-admissible for n, then S
satisfies the generalized condition C and conditions D, E, F and G for n.
We end this chapter with another corollary of Theorem 3.1 that generalizes
Theorem 2.1 from [13].
COROLLARY
3.14. Let L = {ieZ\li ra+l}; with the usual ordering and
let E be a set with n elements. Assume that {6i : Z E | i £ L} is an admissible
array on n symbols. Write Bi = {0i(j) | j Z} and pi = |i?i|, the period of 6i.
Assume that Bi Pi Bi+i 7^ 0 for 1 i m. Let p\ and P2 be positive integers
with pi 1 and write p = p\P2- Suppose that for i G Si C L, gcd(pi_i,pi)|/Oi
and gcd(pi,pi+i\p\. Fori e L\Si, assume that gcd(pi-i,Pi)\p and gcd(pi,pi+i\)p.
Then
ra + 1 P1P2 - S1P2 + si, si := |Si|.
PROOF.
Define r\ = p\, r2 = p = r and S2 = L\S\. Then the conclusion
follows directly from Theorem 3.1.
Previous Page Next Page