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A factorization theorem for lozenge tilings of a hexagon with triangular holes


Authors: M. Ciucu and C. Krattenthaler
Journal: Trans. Amer. Math. Soc. 369 (2017), 3655-3672
MSC (2010): Primary 05A15; Secondary 05A17, 05A19, 05B45, 52C20
DOI: https://doi.org/10.1090/tran/7047
Published electronically: January 6, 2017
MathSciNet review: 3605983
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Abstract: In this paper we present a combinatorial generalization of the fact that the number of plane partitions that fit in a $ 2a\times b\times b$ box is equal to the number of such plane partitions that are symmetric, times the number of such plane partitions for which the transpose is the same as the complement. We use the equivalent phrasing of this identity in terms of symmetry classes of lozenge tilings of a hexagon on the triangular lattice. Our generalization consists of allowing the hexagon to have certain symmetrically placed holes along its horizontal symmetry axis. The special case when there are no holes can be viewed as a new, simpler proof of the enumeration of symmetric plane partitions.


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Additional Information

M. Ciucu
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405-5701

C. Krattenthaler
Affiliation: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Vienna, Austria

DOI: https://doi.org/10.1090/tran/7047
Keywords: Rhombus tilings, lozenge tilings, plane partitions, non-intersecting lattice paths, determinant evaluations
Received by editor(s): April 18, 2014
Received by editor(s) in revised form: October 19, 2015
Published electronically: January 6, 2017
Additional Notes: The first author’s research was partially supported by NSF grants DMS-1101670 and DMS-1501052.
The second author’s research was partially supported by the Austrian Science Foundation FWF, grants Z130-N13 and F50-N15, the latter in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”.
Article copyright: © Copyright 2017 American Mathematical Society