A self-similarity structure generated by King's walk

  • Authors:

  • Marko Razpet

  • Affiliations:

  • Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Jadranska 19, SI-1111 Ljubljana, Slovenia

  • Venue:
  • Discrete Mathematics - Algebraic and topological methods in graph theory
  • Year:
  • 2002

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Abstract

For all non-negative integers i,j let w(i,j) denote the number of all paths in the plane from (0,0) to (i,j) with steps (1,0), (0, 1), and (1, 1). The numbers w(i,j) are known as the Delannoy numbers. They were studied by several authors. Let p be an odd prime and let w(i,j) denote the remainders of w(i,j) when divided by p where 0 ≤ w(i,j) p. The Lucas property of w(i,j) which implies the self-similar pattern of w(i,j) is derived in a new way.The aim of this article is to show that for the array w(i,j) a principal cell exists and that this principal cell has some symmetry properties with special features for p=3,5,7,11,19. Triples of equal numbers of the principal cell are also discussed.