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Transactions of the American Mathematical Society

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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 | References | Similar Articles | Additional Information

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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  • [1] George E. Andrews, Plane partitions. I. The MacMahon conjecture, Studies in foundations and combinatorics, Adv. in Math. Suppl. Stud., vol. 1, Academic Press, New York-London, 1978, pp. 131-150. MR 520557
  • [2] George E. Andrews, Plane partitions. V. The TSSCPP conjecture, J. Combin. Theory Ser. A 66 (1994), no. 1, 28-39. MR 1273289, https://doi.org/10.1016/0097-3165(94)90048-5
  • [3] David M. Bressoud, Proofs and confirmations: The story of the alternating sign matrix conjecture, MAA Spectrum, Mathematical Association of America, Washington, DC; Cambridge University Press, Cambridge, 1999. MR 1718370
  • [4] Mihai Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97. MR 1426739, https://doi.org/10.1006/jcta.1996.2725
  • [5] Mihai Ciucu, The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions, J. Combin. Theory Ser. A 86 (1999), no. 2, 382-389. MR 1685538, https://doi.org/10.1006/jcta.1998.2922
  • [6] M. Ciucu and C. Krattenthaler, The number of centered lozenge tilings of a symmetric hexagon, J. Combin. Theory Ser. A 86 (1999), no. 1, 103-126. MR 1682965, https://doi.org/10.1006/jcta.1998.2918
  • [7] Mihai Ciucu and Christian Krattenthaler, Plane partitions. II. $ 5\frac 12$ symmetry classes, Combinatorial methods in representation theory (Kyoto, 1998) Adv. Stud. Pure Math., vol. 28, Kinokuniya, Tokyo, 2000, pp. 81-101. MR 1855591
  • [8] M. Ciucu and C. Krattenthaler, A factorization theorem for classical group characters, with applications to plane partitions and rhombus tilings, Advances in combinatorial mathematics, Springer, Berlin, 2009, pp. 39-59. MR 2683226, https://doi.org/10.1007/978-3-642-03562-3_3
  • [9] Guy David and Carlos Tomei, The problem of the calissons, Amer. Math. Monthly 96 (1989), no. 5, 429-431. MR 994034, https://doi.org/10.2307/2325150
  • [10] I. M. Gessel and X. Viennot, Determinants, paths, and plane partitions, preprint, 1989; available at http://www.cs.brandeis.edu/˜ira.
  • [11] Basil Gordon, Notes on plane partitions. V, J. Combinatorial Theory Ser. B 11 (1971), 157-168. MR 0280455
  • [12] A. Kasraoui and C. Krattenthaler, Enumeration of symmetric centered rhombus tilings of a hexagon, preprint, 2013; available at arxiv.org/abs/1306.1403.
  • [13] Christoph Koutschan, Manuel Kauers, and Doron Zeilberger, Proof of George Andrews's and David Robbins's $ q$-TSPP conjecture, Proc. Natl. Acad. Sci. USA 108 (2011), no. 6, 2196-2199. MR 2775659, https://doi.org/10.1073/pnas.1019186108
  • [14] Eric H. Kuo, Applications of graphical condensation for enumerating matchings and tilings, Theoret. Comput. Sci. 319 (2004), no. 1-3, 29-57. MR 2074946, https://doi.org/10.1016/j.tcs.2004.02.022
  • [15] Greg Kuperberg, Four symmetry classes of plane partitions under one roof, J. Combin. Theory Ser. A 75 (1996), no. 2, 295-315. MR 1401005, https://doi.org/10.1006/jcta.1996.0079
  • [16] Greg Kuperberg, An exploration of the permanent-determinant method, Electron. J. Combin. 5 (1998), Research Paper 46, 34 pp. (electronic). MR 1663576
  • [17] Bernt Lindström, On the vector representations of induced matroids, Bull. London Math. Soc. 5 (1973), 85-90. MR 0335313
  • [18] Percy A. MacMahon, Combinatory analysis, Two volumes (bound as one), Chelsea Publishing Co., New York, 1960. MR 0141605
  • [19] Soichi Okada, On the generating functions for certain classes of plane partitions, J. Combin. Theory Ser. A 51 (1989), no. 1, 1-23. MR 993645, https://doi.org/10.1016/0097-3165(89)90073-3
  • [20] Robert A. Proctor, Odd symplectic groups, Invent. Math. 92 (1988), no. 2, 307-332. MR 936084, https://doi.org/10.1007/BF01404455
  • [21] Richard P. Stanley, Symmetries of plane partitions, J. Combin. Theory Ser. A 43 (1986), no. 1, 103-113. MR 859302, https://doi.org/10.1016/0097-3165(86)90028-2
  • [22] John R. Stembridge, Nonintersecting paths, Pfaffians, and plane partitions, Adv. Math. 83 (1990), no. 1, 96-131. MR 1069389, https://doi.org/10.1016/0001-8708(90)90070-4
  • [23] John R. Stembridge, The enumeration of totally symmetric plane partitions, Adv. Math. 111 (1995), no. 2, 227-243. MR 1318529, https://doi.org/10.1006/aima.1995.1023

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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

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