On constants in the Füredi--Hajnal and the Stanley--Wilf conjecture

  • Authors:

  • Josef Cibulka

  • Affiliations:

  • Department of Applied Mathematics and Institute for Theoretical Computer Science, Charles University, Malostranské nám. 25, 118 00 Prague, Czech Republic

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2009

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Abstract

For a given permutation matrix P, let f"P(n) be the maximum number of 1-entries in an nxn(0,1)-matrix avoiding P and let S"P(n) be the set of all nxn permutation matrices avoiding P. The Furedi-Hajnal conjecture asserts that c"P:=lim"n"-"~f"P(n)/n is finite, while the Stanley-Wilf conjecture asserts that s"P:=lim"n"-"~|S"P(n)|n is finite. In 2004, Marcus and Tardos proved the Furedi-Hajnal conjecture, which together with the reduction introduced by Klazar in 2000 proves the Stanley-Wilf conjecture. We focus on the values of the Stanley-Wilf limit (s"P) and the Furedi-Hajnal limit (c"P). We improve the reduction and obtain s"P=