The enumeration of prudent polygons by area and its unusual asymptotics

  • Authors:

  • Nicholas R. Beaton;Philippe Flajolet;Anthony J. Guttmann

  • Affiliations:

  • ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia;Algorithms Project, INRIA-Rocquencourt, 78153 Le Chesnay, France;ARC Centre of Excellence for Mathematics and Statistics of Complex Systems, Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2011

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Abstract

Prudent walks are special self-avoiding walks that never take a step towards an already occupied site, and k-sided prudent walks (with k=1,2,3,4) are, in essence, only allowed to grow along k directions. Prudent polygons are prudent walks that return to a point adjacent to their starting point. Prudent walks and polygons have recently been enumerated by length and perimeter by Bousquet-Melou and Schwerdtfeger. We consider the enumeration of prudent polygons by area. For the 3-sided variety, we find that the generating function is expressed in terms of a q-hypergeometric function, with an accumulation of poles towards the dominant singularity. This expression reveals an unusual asymptotic structure of the number of polygons of area n, where the critical exponent is the transcendental number log"23 and the amplitude involves tiny oscillations. Based on numerical data, we also expect similar phenomena to occur for 4-sided polygons. The asymptotic methodology involves an original combination of Mellin transform techniques and singularity analysis, which is of potential interest in a number of other asymptotic enumeration problems.