(-1)-enumeration of plane partitions with complementation symmetry

  • Authors:

  • Theresia Eisenkölbl

  • Affiliations:

  • Institut für Mathematik der Universität Wien, Strudlhofgasse 4, A-1090 Wien, Austria

  • Venue:
  • Advances in Applied Mathematics - Special issue on: Formal power series and algebraic combinatorics in memory of Rodica Simion, 1955-2000
  • Year:
  • 2003

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Abstract

We compute the weighted enumeration of plane partitions contained in a given box with complementation symmetry where adding one half of an orbit of cubes and removing the other half of the orbit changes the weight by -1 as proposed by Kuperberg in [Electron. J. Combin. 5 (1998) R46, pp. 25, 26]. We use nonintersecting lattice path families to accomplish this for transpose-complementary, cyclically symmetric transpose-complementary and totally symmetric self-complementary plane partitions. For symmetric transpose-complementary and self-complementary plane partitions we get partial results. We also describe Kuperberg's proof for the case of cyclically symmetric self-complementary plane partitions.