an enumeration of the sets of non-interfering arithmetic progressions with specified periods英文.pdf

JOURNAL OF COMBINATORIAL THEORY 2, 266-274 (1967)
An Enumeration of the Sets of Non-interfering Arithmetic
Progressions with Specified Periods*
EDWARD A. FAY
U. S. Naval Ordnance Test Station,
Statistics Branch, China Lake, California 93555
Communicated by Marshall Hall, Jr.
ABSTRACT
Let a positive integer n and an ordered n-tuple b = {ba, b~ ..... b~} of positive in-
tegers be given. Let Go be the set of all bib2 ... bn possible choices of an ordered n-tuple
a = {al,as ..... an} of integers such that 0 ~< ai < b~, i = 1, 2 ..... n. Let T be the
subset of Go consisting of those a satisfying the additional conditions
ai ~z~ ay mod(bi, b~); i, ] -- 1, 2 ..... n, i < j;
where (b~, b~) is the greatest common divisor of b~ and b~. Concepts from the most
elementary portions of number theory, group theory, combinatorial analysis, and
graph theory are used to develop a rule for determining the size of T.
Let n be a fixed positive integer, let b = {bl, b2, ..., bn} be a fixed
ordered n-tuple of positive integers, and let a = {al, a2,..., an} be a
variable o