Symmetries of shamrocks IV: The self-complementary case

Ciucu, Mihai

doi:10.1090/proc/15149

Symmetries of shamrocks IV: The self-complementary case
HTML articles powered by AMS MathViewer

by Mihai Ciucu PDF

Proc. Amer. Math. Soc. 149 (2021), 935-951 Request permission

Abstract:

In this paper we enumerate the centrally symmetric lozenge tilings of a hexagon with a shamrock removed from its center. Our proof is based on a variant of Kuo’s graphical condensation method in which only three of the four involved vertices are on the same face. As a special case, we obtain a new proof of the enumeration of the self-complementary plane partitions.

References

George E. Andrews, Plane partitions. III. The weak Macdonald conjecture, Invent. Math. 53 (1979), no. 3, 193–225. MR 549398, DOI 10.1007/BF01389763
David M. Bressoud, Proofs and confirmations, MAA Spectrum, Mathematical Association of America, Washington, DC; Cambridge University Press, Cambridge, 1999. The story of the alternating sign matrix conjecture. MR 1718370
M. Ciucu, Plane partitions I: a generalization of MacMahon’s formula, Mem. Amer. Math. Soc. 178 (2005), no. 839, 107–144.
Mihai Ciucu and Ilse Fischer, A triangular gap of side 2 in a sea of dimers in a $60^\circ$ angle, J. Phys. A 45 (2012), no. 49, 494011, 17. MR 2999729, DOI 10.1088/1751-8113/45/49/494011
Mihai Ciucu and Christian Krattenthaler, A dual of MacMahon’s theorem on plane partitions, Proc. Natl. Acad. Sci. USA 110 (2013), no. 12, 4518–4523. MR 3047654, DOI 10.1073/pnas.1217100110
Mihai Ciucu and Tri Lai, Proof of Blum’s conjecture on hexagonal dungeons, J. Combin. Theory Ser. A 125 (2014), 273–305. MR 3207476, DOI 10.1016/j.jcta.2014.03.008
Mihai Ciucu, The other dual of MacMahon’s theorem on plane partitions, Adv. Math. 306 (2017), 427–450. MR 3581307, DOI 10.1016/j.aim.2016.10.008
Mihai Ciucu and Ilse Fischer, Proof of two conjectures of Ciucu and Krattenthaler on the enumeration of lozenge tilings of hexagons with cut off corners, J. Combin. Theory Ser. A 133 (2015), 228–250. MR 3325633, DOI 10.1016/j.jcta.2015.02.008
Mihai Ciucu, A generalization of Kuo condensation, J. Combin. Theory Ser. A 134 (2015), 221–241. MR 3345305, DOI 10.1016/j.jcta.2015.03.009
Mihai Ciucu and Ilse Fischer, Lozenge tilings of hexagons with arbitrary dents, Adv. in Appl. Math. 73 (2016), 1–22. MR 3433498, DOI 10.1016/j.aam.2015.09.008
Mihai Ciucu, Symmetries of shamrocks, Part I, J. Combin. Theory Ser. A 155 (2018), 376–397. MR 3741434, DOI 10.1016/j.jcta.2017.11.013
Mihai Ciucu, Symmetries of shamrocks II: Axial shamrocks, Electron. J. Combin. 25 (2018), no. 2, Paper No. 2.36, 21. MR 3814270
Mihai Ciucu, Correlation of a macroscopic dent in a wedge with mixed boundary conditions, Trans. Amer. Math. Soc. 373 (2020), no. 3, 2173–2190. MR 4068294, DOI 10.1090/tran/7963
Guy David and Carlos Tomei, The problem of the calissons, Amer. Math. Monthly 96 (1989), no. 5, 429–431. MR 994034, DOI 10.2307/2325150
P. W. Kasteleyn, The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209–1225.
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, DOI 10.1073/pnas.1019186108
Eric H. Kuo, Applications of graphical condensation for enumerating matchings and tilings, Theoret. Comput. Sci. 319 (2004), no. 1-3, 29–57. MR 2074946, DOI 10.1016/j.tcs.2004.02.022
E. H. Kuo, Graphical condensation generalizations involving Pfaffians and determinants, arxiv preprint math:CO/06055154, 2006.
Greg Kuperberg, Symmetries of plane partitions and the permanent-determinant method, J. Combin. Theory Ser. A 68 (1994), no. 1, 115–151. MR 1295786, DOI 10.1016/0097-3165(94)90094-9
Greg Kuperberg, An exploration of the permanent-determinant method, Electron. J. Combin. 5 (1998), Research Paper 46, 34. MR 1663576
Percy A. MacMahon, Combinatory analysis, Chelsea Publishing Co., New York, 1960. Two volumes (bound as one). MR 0141605
Richard P. Stanley, Symmetries of plane partitions, J. Combin. Theory Ser. A 43 (1986), no. 1, 103–113. MR 859302, DOI 10.1016/0097-3165(86)90028-2
John R. Stembridge, Nonintersecting paths, Pfaffians, and plane partitions, Adv. Math. 83 (1990), no. 1, 96–131. MR 1069389, DOI 10.1016/0001-8708(90)90070-4
H. N. V. Temperley and Michael E. Fisher, Dimer problem in statistical mechanics—an exact result, Philos. Mag. (8) 6 (1961), 1061–1063. MR 136398