On the number of frames in binary words



: Harju T, Karki T

Publisher: ELSEVIER SCIENCE BV

: 2011

: Theoretical Computer Science

: THEORETICAL COMPUTER SCIENCE

: THEOR COMPUT SCI

: 39

: 412

: 39

: 5276

: 5284

: 9

: 0304-3975

DOI: https://doi.org/10.1016/j.tcs.2011.05.032(external)


A frame is a square uu, where u is an unbordered word. Let F(n) denote the maximum number of distinct frames in a binary word of length n. We count this number for small values of n and show that F(n) is at most left perpendicularn/2right perpendicular 8 for all n and greater than 7n/30 - epsilon for any positive epsilon and infinitely many n. We also show that Fibonacci words, which are known to contain plenty of distinct squares, have only a few frames. Moreover, by modifying the Thue-Morse word, we prove that the minimum number of occurrences of frames in a word of length n is inverted right perpendicularn/2inverted left perpendicular - 2. (C) 2011 Elsevier B.V. All rights reserved.



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