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Symmetries of plane partitions

1986
*
Journal of combinatorial theory. Series A
*

This operation was suggested by work

doi:10.1016/0097-3165(86)90028-2
fatcat:dfybbs3vqjb7da6ibjsdxibhce
*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.) ...##
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(-1)-enumeration of plane partitions with complementation symmetry
[article]

2002
*
arXiv
*
pre-print

We compute the weighted enumeration

arXiv:math/0011175v3
fatcat:5yixbvhiefhxnpniohqyubnrny
*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. ...##
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Four symmetry classes of plane partitions under one roof
[article]

1995
*
arXiv
*
pre-print

each

arXiv:math/9506225v1
fatcat:uec7hav7lbfonlsjd7slc3sodu
*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*. ...##
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Symmetries of plane partitions and the permanent-determinant method
[article]

1994
*
arXiv
*
pre-print

A similar row-reduction process can be carried out in many

arXiv:math/9410224v1
fatcat:il2gkcxjyffifizvsodyb77lha
*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. ...##
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Enumeration of a symmetry class of plane partitions

1987
*
Discrete Mathematics
*

We give analogous results for other classes

doi:10.1016/0012-365x(87)90165-8
fatcat:ypy5km4uhbcn5bpjzwesfr2drm
*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. ...##
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(−1)-enumeration of plane partitions with complementation symmetry

2003
*
Advances in Applied Mathematics
*

We compute the weighted enumeration

doi:10.1016/s0196-8858(02)00525-0
fatcat:hqi2dpbirza7dnilnlutbiuws4
*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*. ...##
###
Cyclic Sieving for Plane Partitions and Symmetry

2020
*
Symmetry, Integrability and Geometry: Methods and Applications
*

We extend Rhoades's result by also considering

doi:10.3842/sigma.2020.130
fatcat:fxhnolyw2vhyhdf57sxtrlufxe
*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. ...##
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Four Symmetry Classes of Plane Partitions under One Roof

1996
*
Journal of combinatorial theory. Series A
*

each

doi:10.1006/jcta.1996.0079
fatcat:6ysmb4nncnhufchw6szmhaitrm
*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*. ...##
###
Symmetries of plane partitions and the permanent—determinant method

1994
*
Journal of combinatorial theory. Series A
*

A 43, 103-113) gives formulas for the number

doi:10.1016/0097-3165(94)90094-9
fatcat:mzq5iu4vbraste5ogks256xsly
*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. ...##
###
Refined Cauchy and Littlewood identities, plane partitions and symmetry classes of alternating sign matrices

2016
*
Journal of combinatorial theory. Series A
*

The left hand side

doi:10.1016/j.jcta.2015.08.007
fatcat:ofc42ctz2fef5orri6kryid5ie
*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 ...##
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Open boundary quantum Knizhnik–Zamolodchikov equation and the weighted enumeration of plane partitions with symmetries

2007
*
Journal of Statistical Mechanics: Theory and Experiment
*

We propose new conjectures relating sum rules for the polynomial solution

doi:10.1088/1742-5468/2007/01/p01024
fatcat:uuma4lzcwreopcidp5fbakqhii
*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. ...##
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Alternating sign matrices with reflective symmetry and plane partitions: n+3 pairs of equivalent statistics
[article]

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*. ...

##
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Some hidden relations involving the ten symmetry classes of plane partitions

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 . ...

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

2015
*
arXiv
*
pre-print

These notes provide a survey

arXiv:1503.05934v2
fatcat:ls5qfjsu5vh3hbghxfsy3a4bl4
*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 ...##
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Mirror Symmetry in Flavored Affine D-type Quivers
[article]

2014
*
arXiv
*
pre-print

We present non-trivial checks

arXiv:1401.6462v1
fatcat:ixlgsdrxhbhbhnbdjbitizn3ma
*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. ...
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