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Symmetries of plane partitions

Richard P Stanley
1986 Journal of combinatorial theory. Series A  
This operation was suggested by work of Mills, Robbins, and Rumsey. There then arise a total of ten inequivalent problems concerned with the enumeration of plane partitions with a given symmetry.  ...  We introduce a new symmetry operation, called complementation, on plane partitions whose three-dimensional diagram is contained in a given box.  ...  MacMahon defined six symmetry operations on plane partitions and raised the problem of enumerating plane partitions with given symmetries. (Precise statements and references are given below.)  ... 
doi:10.1016/0097-3165(86)90028-2 fatcat:dfybbs3vqjb7da6ibjsdxibhce

(-1)-enumeration of plane partitions with complementation symmetry [article]

Theresia Eisenkölbl
2002 arXiv   pre-print
We compute the weighted enumeration of plane partitions contained in a given box with complementation symmetry where adding one half of an orbit of cubes and removing the other half of the orbit changes  ...  We also describe Kuperberg's proof for the case of cyclically symmetric self-complementary plane partitions.  ...  A plane partition can have several kinds of symmetries which we now list.  ... 
arXiv:math/0011175v3 fatcat:5yixbvhiefhxnpniohqyubnrny

Four symmetry classes of plane partitions under one roof [article]

Greg Kuperberg
1995 arXiv   pre-print
each of the ten symmetry classes of plane partitions.  ...  After a cosmetic generalization of the Kasteleyn method, we identify the matrices in the four determinantal cases (plain plane partitions, cyclically symmetric plane partitions, transpose-complement plane  ...  There are three natural symmetry operations on plane partitions: 1. Transposition, τ, is a reflection through a diagonal plane of the box of a plane partition.  ... 
arXiv:math/9506225v1 fatcat:uec7hav7lbfonlsjd7slc3sodu

Symmetries of plane partitions and the permanent-determinant method [article]

Greg Kuperberg
1994 arXiv   pre-print
A similar row-reduction process can be carried out in many of the other cases, and we analyze three other symmetry classes of plane partitions for comparison.  ...  A 43 (1986), 103--113], Stanley gives formulas for the number of plane partitions in each of ten symmetry classes. This paper together with results by Andrews [J. Combin. Theory Ser.  ...  Introduction In the paper [13] , Richard Stanley describes ten symmetry classes of plane partitions and gives formulas for their enumeration.  ... 
arXiv:math/9410224v1 fatcat:il2gkcxjyffifizvsodyb77lha

Enumeration of a symmetry class of plane partitions

W.H Mills, David P Robbins, Howard Rumsey
1987 Discrete Mathematics  
We give analogous results for other classes of plane partitions including descending plane partitions.  ...  In this paper we show that the number of tc-symmetric plane partitions in ~2n is given by We show that, with a suitable assignment of weights, the generating function for tc-symmetric plane partitions  ...  These tc-symmetric plane partitions are one of the ten symmetry classes of plane partitions described in [6] . For all ten cases the numbers of plane partitions seem to be given by nice formulas.  ... 
doi:10.1016/0012-365x(87)90165-8 fatcat:ypy5km4uhbcn5bpjzwesfr2drm

(−1)-enumeration of plane partitions with complementation symmetry

Theresia Eisenkölbl
2003 Advances in Applied Mathematics  
We compute the weighted enumeration of plane partitions contained in a given box with complementation symmetry where adding one half of an orbit of cubes and removing the other half of the orbit changes  ...  We also describe Kuperberg's proof for the case of cyclically symmetric self-complementary plane partitions.  ...  Acknowledgment We thank Greg Kuperberg for his permission to reproduce his proof of Theorem 7, the (−1)-enumeration of cyclically symmetric self-complementary plane partitions.  ... 
doi:10.1016/s0196-8858(02)00525-0 fatcat:hqi2dpbirza7dnilnlutbiuws4

Cyclic Sieving for Plane Partitions and Symmetry

Sam Hopkins, University of Minnesota, USA
2020 Symmetry, Integrability and Geometry: Methods and Applications  
We extend Rhoades's result by also considering symmetries of plane partitions: specifically, complementation and transposition.  ...  Finally, we explain how these symmetry results also apply to the rowmotion operator on plane partitions, which is closely related to promotion.  ...  I benefited from the use of Sage mathematics software [71, 72] in the course of this research.  ... 
doi:10.3842/sigma.2020.130 fatcat:fxhnolyw2vhyhdf57sxtrlufxe

Four Symmetry Classes of Plane Partitions under One Roof

Greg Kuperberg
1996 Journal of combinatorial theory. Series A  
each of the ten symmetry classes of plane partitions.  ...  Stanley [10] and Robbins recognized that there are ten symmetry classes of plane partitions in a box, and, with the aid of computer experiments and other work, that the number of plane partitions in each  ...  There are three natural symmetry operations on plane partitions: 1. Transposition, τ, is a reflection through a diagonal plane of the box of a plane partition.  ... 
doi:10.1006/jcta.1996.0079 fatcat:6ysmb4nncnhufchw6szmhaitrm

Symmetries of plane partitions and the permanent—determinant method

Greg Kuperberg
1994 Journal of combinatorial theory. Series A  
A 43, 103-113) gives formulas for the number of plane partitions in each of 10 symmetry classes. This paper together with papers of G. Andrews (J. Combin. Theory Ser. A, to appear) and J.  ...  A similar row-reduction process can be carried out in many of the other cases, and we analyze three other symmetry classes of plane partitions for comparison. [0, a] × [0, b] × [0, c].  ...  INTRODUCTION Richard Stanley [12] describes 10 symmetry classes of plane partitions and gives formulas for their enumeration.  ... 
doi:10.1016/0097-3165(94)90094-9 fatcat:mzq5iu4vbraste5ogks256xsly

Refined Cauchy and Littlewood identities, plane partitions and symmetry classes of alternating sign matrices

D. Betea, M. Wheeler
2016 Journal of combinatorial theory. Series A  
The left hand side of each of our identities is a simple refinement of a relevant Cauchy or Littlewood identity, allowing them to be interpreted as generating series for plane partitions.  ...  We prove and conjecture some new symmetric function identities, which equate the generating series of 1. Plane partitions, subject to certain restrictions and weightings, and 2.  ...  Acknowledgments We would like to thank Andrea Sportiello for suggesting the idea of extending the identity (25) to ASM symmetry classes, which strongly motivated this work; Paul Zinn-Justin for explaining  ... 
doi:10.1016/j.jcta.2015.08.007 fatcat:ofc42ctz2fef5orri6kryid5ie

Open boundary quantum Knizhnik–Zamolodchikov equation and the weighted enumeration of plane partitions with symmetries

P Di Francesco
2007 Journal of Statistical Mechanics: Theory and Experiment  
We propose new conjectures relating sum rules for the polynomial solution of the qKZ equation with open (reflecting) boundaries as a function of the quantum parameter q and the τ-enumeration of Plane Partitions  ...  We also find a conjectural relation à la Razumov-Stroganov between the τ→ 0 limit of the qKZ solution and refined numbers of Totally Symmetric Self Complementary Plane Partitions.  ...  Plane Partitions with symmetries In this section, we recall a few know facts on Plane Partitions with symmetries, related in particular to their expression as NILP and to their explicit enumeration.  ... 
doi:10.1088/1742-5468/2007/01/p01024 fatcat:uuma4lzcwreopcidp5fbakqhii

Alternating sign matrices with reflective symmetry and plane partitions: n+3 pairs of equivalent statistics [article]

Ilse Fischer, Hans Höngesberg
2022 arXiv   pre-print
of size 2 that exhibit a cyclical as well as a vertical symmetry, but no bijection between these two classes of objects has been constructed so far.  ...  The equinumeracy of VSASMs and the lozenge tilings is then an easy consequence of this result, which is obtained by specializing the generating functions to signed enumerations for both types of objects  ...  pairs of plane partitions.  ... 
arXiv:2207.04469v1 fatcat:xzibhuf5tngqdhrur43u3777gi

Some hidden relations involving the ten symmetry classes of plane partitions

John R Stembridge
1994 Journal of combinatorial theory. Series A  
PLANE PARTITIONS 373 JOHN R. STEMBRIDGE could be found, it would substantially reduce the number of "miraculous" closed formulas one is forced to accept.  ...  In Section 3, we treat the case of cyclic symmetry. Although this is still a matter of repackaging, the details are somewhat more delicate.  ...  The plane partitions invariant under S; are those whose transpose coincides with their complement; the symmetry class ST consists of those plane partitions invariant under both C, and Sf .  ... 
doi:10.1016/0097-3165(94)90112-0 fatcat:s5n6w6irjfb6zdlqufpyadcuj4

Plane partitions in the work of Richard Stanley and his school [article]

C. Krattenthaler
2015 arXiv   pre-print
These notes provide a survey of the theory of plane partitions, seen through the glasses of the work of Richard Stanley and his school.  ...  I start by defining the symmetry operations which give rise to 10 symmetry classes of plane partitions.  ...  The 10 symmetry classes of plane partitions The purpose of this section is to summarise the programme of enumeration of symmetry classes of plane partitions, the history of which was reviewed in Section  ... 
arXiv:1503.05934v2 fatcat:ls5qfjsu5vh3hbghxfsy3a4bl4

Mirror Symmetry in Flavored Affine D-type Quivers [article]

Anindya Dey
2014 arXiv   pre-print
We present non-trivial checks of three dimensional mirror symmetry for N=4, D̂_N quiver gauge theories with unitary gauge groups using partition function on a round sphere.  ...  5-planes respectively, in addition to NS5 and D5 branes.  ...  Similarly, an orbifold 5-plane is S-dual to a O5 0 planes (i.e. O5 − plane coincident with a D5) -this plays a crucial role in mirror symmetry for affine D-type quivers. S 3 partition function.  ... 
arXiv:1401.6462v1 fatcat:ixlgsdrxhbhbhnbdjbitizn3ma
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