Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Four factorization formulas for plane partitions


Author: Mihai Ciucu
Journal: Proc. Amer. Math. Soc. 144 (2016), 1841-1856
MSC (2010): Primary 05A15, 05A17; Secondary 05A19
DOI: https://doi.org/10.1090/proc/12800
Published electronically: January 20, 2016
MathSciNet review: 3460147
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: All ten symmetry classes of plane partitions that fit in a given box are known to be enumerated by simple product formulas, but there is still no unified proof for all of them. Progress towards this goal can be made by establishing identities connecting the various symmetry classes. We present in this paper four such identities, involving all ten symmetry classes. We discuss their proofs and generalizations. The main result of this paper is to give a generalization of one of them, in the style of the identity presented in ``A factorization theorem for rhombus tilings,'' M. Ciucu and C. Krattenthaler, arXiv:1403.3323.


References [Enhancements On Off] (What's this?)

  • [1] George E. Andrews, Plane partitions. V. The TSSCPP conjecture, J. Combin. Theory Ser. A 66 (1994), no. 1, 28-39. MR 1273289 (95g:05010), https://doi.org/10.1016/0097-3165(94)90048-5
  • [2] Mihai Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theory Ser. A 77 (1997), no. 1, 67-97. MR 1426739 (98a:05112), https://doi.org/10.1006/jcta.1996.2725
  • [3] 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 (2000b:05016), https://doi.org/10.1006/jcta.1998.2922
  • [4] M. Ciucu, Plane partitions I: A generalization of MacMahon's formula, Mem. Amer. Math. Soc. 178 (2005), no. 839, 107-144.
  • [5] Mihai Ciucu, Lozenge tilings with gaps in a $ 90^\circ $ wedge domain with mixed boundary conditions, Comm. Math. Phys. 334 (2015), no. 1, 507-532. MR 3304284, https://doi.org/10.1007/s00220-014-2138-2
  • [6] M. Ciucu and C. Krattenthaler, A factorization theorem for rhombus tilings, preprint (2014), arXiv:1403.3323.
  • [7] P. A. MacMahon, Memoir on the theory of the partition of numbers--Part V. Partitions in two-dimensional space, Phil. Trans. R. S., 1911, A.
  • [8] Guy David and Carlos Tomei, The problem of the calissons, Amer. Math. Monthly 96 (1989), no. 5, 429-431. MR 994034 (90c:51024), https://doi.org/10.2307/2325150
  • [9] Greg Kuperberg, Symmetries of plane partitions and the permanent-determinant method, J. Combin. Theory Ser. A 68 (1994), no. 1, 115-151. MR 1295786 (96b:05009), https://doi.org/10.1016/0097-3165(94)90094-9
  • [10] Richard P. Stanley, Symmetries of plane partitions, J. Combin. Theory Ser. A 43 (1986), no. 1, 103-113. MR 859302 (87m:05017a), https://doi.org/10.1016/0097-3165(86)90028-2
  • [11] John R. Stembridge, The enumeration of totally symmetric plane partitions, Adv. Math. 111 (1995), no. 2, 227-243. MR 1318529 (96b:05010), https://doi.org/10.1006/aima.1995.1023

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 05A15, 05A17, 05A19

Retrieve articles in all journals with MSC (2010): 05A15, 05A17, 05A19


Additional Information

Mihai Ciucu
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405

DOI: https://doi.org/10.1090/proc/12800
Received by editor(s): October 16, 2014
Published electronically: January 20, 2016
Additional Notes: This research was supported in part by NSF grant DMS-1101670
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society