Concrete mathematics : a foundation for computer science

140 NUMBER THEORY
The problem of counting these configurations was first solved by P. A. Mac-
Mahon in 1892 [212].
There’s no obvious recurrence for N (m, n), but we can count the necklaces
by breaking them each into linear strings in m ways and considering the
resulting fragments. For example, when m = 4 and n = 2 we get
RRRR RRRR RRRR RRRR
RRBR RRRB BRRR RBRR
RBBR RRBB BRRB BBRR
RBRB BRBR RBRB BRBR
RBBB BRBB BBRB BBBR
BBBB BBBB BBBB BBBB
Each of the nm possible patterns appears at least once in this array of
mN(m,n) strings, and some patterns appear more than once. How many
times does a pattern a~. . . a,,-, appear? That’s easy: It’s the number of
cyclic shifts ok . . . a,-, a0 . . . ok-1 that produce the same pattern as the original
a0 . . . a,-, . For example, BRBR occurs twice, because the four ways to
cut the necklace formed from BRBR produce four cyclic shifts (BRBR, RBRB,
BRBR, RBRB); two of these coincide with BRBR itself. This argument shows
that
mN(m,n) = t
q,,...,a,e,ES, O$k