Normal, Abby Normal, Prefix Normal

A prefix normal word is a binary word with the property thatno substring has more 1s than the prefix of the same length. This class of words is important in the context of binary jumbled pattern matching. In this paper we present results about the number pnw(n) of prefix normal words of length n, showing that pnw(n) = Omega(2^{n−c n ln n}) for some c, and pnw(n) = O(2^n (ln n)^2/2) . We introduce efficient algorithms for testing the prefix normal property and a “mechanical algorithm” for computing prefix normal forms. We also include games which can be played with prefix normal words. In these games Alice wishes to stay normal but Bob wants to drive her “abnormal” – we discuss which parameter settings allow Alice to succeed.