Counting lattice paths taking steps in infinitely many directions under special access restrictions

  • Authors:

  • Katherine Humphreys;Heinrich Niederhausen

  • Affiliations:

  • Department of Mathematical Science, Florida Atlantic University, Boca Raton, FL;Department of Mathematical Science, Florida Atlantic University, Boca Raton, FL

  • Venue:
  • Theoretical Computer Science - Combinatorics of the discrete plane and tilings
  • Year:
  • 2004

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Abstract

We count lattice paths that are confined to the first quadrant by the nature of their step vectors. If no further restrictions apply, a path can go from any point to infinitely many others, but each point on the path has only finitely many predecessors. By "further restrictions" we mean a boundary line above which the paths may have to stay. Access privilege to the boundary line itself is granted from certain lattice points in the form of a special access step set, which itself may be infinite. We also count the number of paths that contact the weak boundary a given number of times. We approach explicit solutions of such enumeration problems via Sheffer polynomials and functionals, using results of the Umbral Calculus.