Signed lozenge tilings

D. Cook, Uwe Nagel

Research output: Contribution to journalArticlepeer-review

4 Citations (SciVal)


It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing tri- angles. We call the resulting shapes triangular regions. We establish signed versions of the latter three bijections for triangular regions. We first investigate the tileability of triangular regions by lozenges. Then we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. Using a new method that we call resolution of a puncture, we show that the two signs are in fact equivalent. As a consequence, we obtain the equality of determinants, up to sign, that enumerate signed perfect matchings and signed families of lattice paths of a triangular region, respectively. We also describe triangular regions for which the signed enumerations agree with the unsigned enumerations.

Original languageEnglish
Article number#P1.9
JournalElectronic Journal of Combinatorics
Issue number1
StatePublished - Jan 20 2017

Bibliographical note

Publisher Copyright:
© 2017, Australian National University. All Rights Reserved.


  • Determinants
  • Lozenge tilings
  • Non-intersecting lattice paths
  • Perfect matchings
  • Punctures

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics


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