A194850 Number of prefix normal words of length n.

2, 3, 5, 8, 14, 23, 41, 70, 125, 218, 395, 697, 1273, 2279, 4185, 7568, 13997, 25500, 47414, 87024, 162456, 299947, 562345, 1043212, 1962589, 3657530, 6900717, 12910042, 24427486, 45850670, 86970163, 163756708, 311283363, 587739559, 1119581278, 2119042830

Offset: 1

Author

Gabriele Fici, Sep 04 2011

Keywords

• nonn

Comments

A binary word of length n is prefix normal if for all 1 <= k <= n, no factor of length k has more a's than the prefix of length k. That is, abbabab is not prefix normal because aba has more a's than abb. - Zsuzsanna Liptak, Oct 12 2011

a(n) <= A062692(n): every prefix normal word is a pre-necklace, but the converse is not true, see the Fici/Lipták reference. - Joerg Arndt, Jul 20 2013

Examples

			For n=3: aaa, aab, abb, aba, bbb are all prefix normal words. - _Zsuzsanna Liptak_, Oct 12 2011
		

Links

• Zsuzsanna Liptak, Table of n, a(n) for n = 1..50
• Paul Balister and Stefanie Gerke, The asymptotic number of prefix normal words, arXiv:1903.07957 [math.CO], 2019.
• P. Burcsi, G. Fici, Zs. Lipták, F. Ruskey, and J. Sawada, On Combinatorial Generation of Prefix Normal Words, arXiv:1401.6346 [cs.DS], 2014.
• P. Burcsi, G. Fici, Z. Lipták, F. Ruskey, and J. Sawada, Normal, Abby Normal, Prefix Normal, arXiv preprint arXiv:1404.2824 [cs.FL], 2014.
• P. Burcsi, G. Fici, Z. Lipták, F. Ruskey, and J. Sawada, On prefix normal words and prefix normal forms, Preprint, 2016.
• Péter Burcsi, Gabriele Fici, Zsuzsanna Lipták, Rajeev Raman, and Joe Sawada, Generating a Gray code for prefix normal words in amortized polylogarithmic time per word, arXiv:2003.03222 [cs.DS], 2020.
• Péter Burcsi, Gabriele Fici, Zsuzsanna Lipták, Frank Ruskey, and Joe Sawada, On Prefix Normal Words and Prefix Normal Forms, arXiv:1611.09017 [cs.DM], 2016.
• Ferdinando Cicalese, Zsuzsanna Lipták, and Massimiliano Rossi, Bubble-Flip—A new generation algorithm for prefix normal words, arXiv:1712.05876 [cs.DS], 2017-2018; Theoretical Computer Science, Volume 743, 26 September 2018, Pages 38-52.
• Ferdinando Cicalese, Zsuzsanna Lipták, and Massimiliano Rossi, On Infinite Prefix Normal Words, arXiv:1811.06273 [math.CO], 2018.
• G. Fici and Zs. Lipták, On Prefix Normal Words
• G. Fici and Zs. Lipták, On Prefix Normal Words, Developments in Language Theory 2011, Lecture Notes in Computer Science 6795, 228-238.
• Pamela Fleischmann, On Special k-Spectra, k-Locality, and Collapsing Prefix Normal Words, Ph.D. Dissertation, Kiel University (Germany, 2021).
• Pamela Fleischmann, Mitja Kulczynski, and Dirk Nowotka, On Collapsing Prefix Normal Words, arXiv:1905.11847 [cs.FL], 2019.
• Pamela Fleischmann, Mitja Kulczynski, Dirk Nowotka, and Danny Bøgsted Poulsen, On Collapsing Prefix Normal Words, Language and Automata Theory and Applications (LATA 2020) LNCS Vol. 12038, Springer, Cham, 412-424.
• Zsuzsanna Lipták, Open problems on prefix normal words, also in Dagstuhl Reports (2018) Vol. 8, Issue 7, 59-61.

Crossrefs

Cf. A062692 (binary pre-necklaces).

See A238109 for a list of the prefix-normal words.

Programs

Python

from itertools import product
def is_prefix_normal(w):
  for k in range(1, len(w)+1):
    weight0 = w[:k].count("1")
    for j in range(1, len(w)-k+1):
      weightj = w[j:j+k].count("1")
      if weightj > weight0: return False
  return True
def a(n):
  return sum(is_prefix_normal(w) for w in product("01", repeat=n))
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Dec 19 2020

Extensions

More terms added by Zsuzsanna Liptak, Oct 12 2011

Further terms added by Zsuzsanna Liptak, Jan 29 2014