272 research outputs found
Hamming Distance for Conjugates
Let x, y be strings of equal length. The Hamming distance h(x,y) between x
and y is the number of positions in which x and y differ. If x is a cyclic
shift of y, we say x and y are conjugates. We consider f(x,y), the Hamming
distance between the conjugates xy and yx. Over a binary alphabet f(x,y) is
always even, and must satisfy a further technical condition. By contrast, over
an alphabet of size 3 or greater, f(x,y) can take any value between 0 and
|x|+|y|, except 1; furthermore, we can always assume that the smaller string
has only one type of letter.Comment: revisio
Polynomial versus Exponential Growth in Repetition-Free Binary Words
It is known that the number of overlap-free binary words of length n grows
polynomially, while the number of cubefree binary words grows exponentially. We
show that the dividing line between polynomial and exponential growth is 7/3.
More precisely, there are only polynomially many binary words of length n that
avoid 7/3-powers, but there are exponentially many binary words of length n
that avoid (7/3+)-powers. This answers an open question of Kobayashi from 1986.Comment: 12 page
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