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### Enumeration of alternating sign matrices of even size (quasi)-invariant under a quarter-turn rotation [article]

Jean-Christophe Aval
2009 arXiv   pre-print
The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn.  ...  The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestricted ASM's and the number of half-turn symmetric ASM's.  ...  Fig. 4 : 4 Partition functions for (q)QTASM of even size Lemma 8 [ 8 Yang-Baxter equation] If xyz = a, We multiply the left-hand side by σ(az), with z = axy.  ...

### Enumeration of Alternating Sign Matrices of Even Size (Quasi-)Invariant under a Quarter-Turn Rotation

Jean-Christophe Aval, Philippe Duchon
2010 Electronic Journal of Combinatorics
The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn.  ...  The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestricted ASM's and the number of half-turn symmetric ASM's.  ...  ASM's quasi-invariant under a quarter-turn The class of ASM's invariant under a rotation by a quarter-turn (QTASM) is non-empty in size 4N − 1, 4N , and 4N + 1.  ...

### Enumeration of alternating sign matrices of even size (quasi)-invariant under a quarter-turn rotation

Jean-Christophe Aval, Philippe Duchon
2009 Discrete Mathematics & Theoretical Computer Science
International audience The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn.  ...  L'objet de ce travail est d'énumérer les matrices à signes alternants (ASM) quasi-invariantes par rotation d'un quart-de-tour.  ...  . , x 2N , x, y) Enumeration of alternating sign matrices of even size (quasi)-invariant under a quarter-turn rotation 119 Theorem 6 When a = ω 6 = exp(iπ/3), one has for N ≥ 1: Z QT (4N ; X 2N −1 ,  ...

### Enumeration of alternating sign matrices of even size (quasi-)nvariant under a quarter-turn rotation [article]

Jean-Christophe Aval
2009 arXiv   pre-print
The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn.  ...  The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestricted ASM's and the number of half-turn symmetric ASM's.  ...  ASM's quasi-invariant under a quarter-turn The class of ASM's invariant under a rotation by a quarter-turn (QTASM) is non-empty in size 4N − 1, 4N , and 4N + 1.  ...

### Enumeration of alternating sign matrices of even size (quasi-)invariant under a quarter-turn rotation

Jean-Christophe Aval, Philippe Duchon
2010 unpublished
The aim of this work is to enumerate alternating sign matrices (ASM) that are quasi-invariant under a quarter-turn.  ...  The enumeration formula (conjectured by Duchon) involves, as a product of three terms, the number of unrestricted ASM's and the number of half-turn symmetric ASM's.  ...  ASM's quasi-invariant under a quarter-turn The class of ASM's invariant under a rotation by a quarter-turn (QTASM) is non-empty in size 4N − 1, 4N, and 4N + 1.  ...

### On the link pattern distribution of quarter-turn symmetric FPL configurations

Philippe Duchon
2008 Discrete Mathematics & Theoretical Computer Science
International audience We present new conjectures on the distribution of link patterns for fully-packed loop (FPL) configurations that are invariant, or almost invariant, under a quarter turn rotation,  ...  As a byproduct, we get a formula for the enumeration of a new class of quasi-symmetry of plane partitions.  ...  comments on a previous version of this paper.  ...

### On the link pattern distribution of quarter-turn symmetric FPL configurations [article]

Philippe Duchon
2007 arXiv   pre-print
We present new conjectures on the distribution of link patterns for fully-packed loop (FPL) configurations that are invariant, or almost invariant, under a quarter turn rotation, extending previous conjectures  ...  As a byproduct, we get a formula for the enumeration of a new class of quasi-symmetry of plane partitions.  ...  QTFPLs of even size N only exist if N is a multiple of 4, which is easiest seen on the corresponding alternating-sign matrices: the sum of entries in any quarter of the square has to be exactly a quarter  ...

### Enumeration of Perfect Matchings in Graphs with Reflective Symmetry

Mihai Ciucu
1997 Journal of combinatorial theory. Series A
Finally, we consider symmetry classes of perfect matchings of the Aztec diamond graph and we solve the previously open problem of enumerating the matchings that are invariant under a rotation by 90 degrees  ...  A plane graph is called symmetric if it is invariant under the re ection across some straight line.  ...  I would also like to thank the referee for the careful reading of the manuscript.  ...

### Twenty-Vertex Model with Domain Wall Boundaries and Domino Tilings

Philippe Di Francesco, Emmanuel Guitter
2020 Electronic Journal of Combinatorics
The four enumeration problems are reformulated in terms of four types of Alternating Phase Matrices with entries \$0\$ and sixth roots of unity, subject to suitable alternation conditions.  ...  In types 1 and 2, the configurations are shown to be equinumerous to the quarter-turn symmetric domino tilings of an Aztec-like holey square, with a central cross-shaped hole.  ...  We wish to enumerate the domino tilings of this domain that are invariant under a quarter-turn rotation (i.e. of angle π/2) around the center of the cross.  ...

### A one-parameter refinement of the Razumov-Stroganov correspondence [article]

Luigi Cantini, Andrea Sportiello
2012 arXiv   pre-print
We introduce and prove a one-parameter refinement of the Razumov-Stroganov correspondence.  ...  We show that the enumeration vector associated to such FPLs, weighted according to the position of the straight line and refined according to the link pattern for the black boundary points, is the ground  ...  This family of domains includes some of the "symmetry classes" of FPL (or equivalently of Alternating Sign Matrices) for which a Razumov-Stroganov conjecture was formulated, namely HTASM and QTASM (half-turn  ...

### Diagonally and antidiagonally symmetric alternating sign matrices of odd order

Roger E. Behrend, Ilse Fischer, Matjaž Konvalinka
We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection  ...  Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved.  ...  The third author acknowledges support from Research Programs L1-069 and Z1-5434 of the Slovenian Research Agency.  ...

### Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order [article]

Arvind Ayyer, Roger E. Behrend, Ilse Fischer
2016 arXiv   pre-print
For each α∈{0,1,-1}, we count alternating sign matrices that are invariant under reflections in the diagonal and in the antidiagonal (DASASMs) of fixed odd order with a maximal number of α's along the  ...  In these enumerations, we encounter product formulas that have previously appeared in plane partition or alternating sign matrix counting, namely for the number of all alternating sign matrices, the number  ...  order and quarter-turn symmetric ASMs of even order.  ...

### Arctic Curves of the Six-Vertex Model on Generic Domains: The Tangent Method

F. Colomo, A. Sportiello
2016 Journal of statistical physics
We adapt the method to work for a large class of domains, and for other models exhibiting limit shape phenomena.  ...  We study in detail some examples, and derive, in particular, the Arctic curve of the six-vertex model in a triangoloid domain at the ice-point.  ...  Alternating Sign Matrices An Alternating Sign Matrix (ASM) of size n is an n × n matrix valued in {0, ±1}, such that: (i) non-zero entries alternate in sign along rows and columns; (ii) the sum of entries  ...

### Arctic curves of the twenty-vertex model with domain wall boundaries [article]

Bryan Debin, Philippe Di Francesco, Emmanuel Guitter
2019 arXiv   pre-print
We finally compute the arctic curve of the Quarter-turn symmetric Holey Aztec Domino Tiling (QTHADT) model, a problem closely related to the 20V model and whose asymptotics may be analyzed via a similar  ...  The latter displays a large variety of shapes depending on the weights and separates a central liquid phase from up to six different frozen phases.  ...  -Vlaanderen (FWO) under EOS project no 30889451.  ...

### Remarks on the construction and the ergodicity properties of dual unitary quantum circuits (with an Appendix by Roland Bacher and Denis Serre) [article]

Márton Borsi, Balázs Pozsgay
2022 arXiv   pre-print
They appear locally as the most chaotic and most scrambling circuits, nevertheless they can show global signs of non-ergodicity: if the perfect tensor is constructed from a linear map over finite fields  ...  Afterwards we consider the ergodicity properties of a special class of dual unitary models, where the local gate is a permutation matrix.  ...  Alternatively, for an even number of legs they can be seen as multi-unitary operators for every bi-partitioning of the legs into two subsets with equal size  .  ...
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