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### Ribbon Tile Invariants from the Signed Area

Cristopher Moore, Igor Pak
2002 Journal of combinatorial theory. Series A
Here we present a complete proof of the conjecture, which works by associating ribbon tiles with certain polygons in the complex plane, and deriving invariants from the signed area of these polygons.  ...  Ribbon tiles are polyominoes consisting of n squares laid out in a path, each step of which goes north or east. Tile invariants were first introduced by the second author (2000, Trans. Amer. Math.  ...  This completes the proof. L f © mod 2=(f 1 +2f 2 + · · · +(m − 1) f m − 1 )+f g mod 2 (cf. NEW RIBBON TILE INVARIANTS AND THE SIGNED AREA Let T n be the set of ribbon tiles, defined as above.  ...

### Exploring the "Rubik's Magic" universe [article]

Maurizio Paolini
2016 arXiv   pre-print
The set of constructible configurations (those that can be obtained by manipulation of the undecorated puzzle from the starting situation) is a subset of the set of configurations with vanishing invariants  ...  parallel to one coordinate plane; and the planar "face-up" shapes, with all tiles (considered of infinitesimal width) lying in a common plane and without superposed consecutive tiles.  ...  In this latter case there is still a "bounce" of the ribbon when it goes from the previous tile to the next (superposed) tile: the ribbon travels from below the lower tile to above the upper tile or viceversa  ...

### Exploring the "Rubik's Magic" Universe

Maurizio Paolini
2017 Recreational Mathematics Magazine
The set of constructible configurations (those that can be obtained by manipulation of the undecorated puzzle from the starting situation) is a subset of the set of configurations with vanishing invariants  ...  parallel to one coordinate plane; and the planar "face-up" shapes, with all tiles (considered of infinitesimal width) lying in a common plane and without superposed consecutive tiles.  ...  In this latter case there is still a "bounce" of the ribbon when it goes from the previous tile to the next (superposed) tile: the ribbon travels from below the lower tile to above the upper tile or viceversa  ...

### Ribbon tile invariants

Igor Pak
2000 Transactions of the American Mathematical Society
We also discuss signed tilings by the ribbon tiles and apply our results to the tileability problem.  ...  We compute the tile counting group for a large set of ribbon tiles, also known as rim hooks, in a context of representation theory of the symmetric group.  ...  Nathan Lulov asked me the initial question which later became Theorem 1.8.  ...

### Signed Tilings by Ribbon L n-Ominoes, n Odd, via Gröbner Bases

Viorel Nitica
2016 Open Journal of Discrete Mathematics
We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n.  ...  A consequence of our technique, based on the exhibition of an explicit Grӧbner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes.  ...  Both of these regions can be tiled by r ribbon tiles of area n as in Figure 11 .  ...

### Signed tilings by ribbon L n-ominoes, n odd, via Groebner bases [article]

Viorel Nitica
2016 arXiv   pre-print
We show that a rectangle can be signed tiled by ribbon L n-ominoes, n odd, if and only if it has a side divisible by n.  ...  A consequence of our technique, based on the exhibition of an explicit Groebner basis, is that any k-inflated copy of the skewed L n-omino has a signed tiling by skewed L n-ominoes.  ...  Acknowledgement The author was partially supported by Simons Foundation Grant 208729. References  ...

### Tile invariants: new horizons

Igor Pak
2003 Theoretical Computer Science
Let T be a ÿnite set of tiles. The group of invariants G(T), introduced by Pak (Trans.  ...  AMS 352 (2000) 5525) , is a group of linear relations between the number of copies of tiles in tilings of the same region.  ...  First, Scott She eld resolved most of the open problems on ribbon tilings in "Ribbon tilings and multidimensional height functions", arXiv preprint math.CO=0107095.  ...

### Page 822 of Mathematical Reviews Vol. , Issue 2003B [page]

2003 Mathematical Reviews
We give certain results for the case k = 4.” 2003b:05049 05B50 Moore, Cristopher (1-NM-C; Albuquerque, NM); Pak, Igor (1-MIT; Cambridge, MA) Ribbon tile invariants from the signed area.  ...  Here we present a complete proof of the conjecture, which works by associating ribbon tiles with certain polygons in the complex plane, and deriving invariants from the signed area of these polygons.”  ...

### The topology of tile invariants

Michael P. Hitchman
2015 Rocky Mountain Journal of Mathematics
In this topological setting we derive some well-known tile invariants, one of which we apply to the solution of a tiling question involving modified rectangles.  ...  from T.  ...  The author would like to thank Bill Bogley for our discussions as this paper took shape, and the reviewers for their helpful comments and suggestions.  ...

### Rational Parking Functions and LLT Polynomials [article]

Eugene Gorsky, Mikhail Mazin
2015 arXiv   pre-print
We prove that the combinatorial side of the "Rational Shuffle Conjecture" provides a Schur-positive symmetric polynomial.  ...  Furthermore, we prove that the contribution of a given rational Dyck path can be computed as a certain skew LLT polynomial, thus generalizing the result of Haglund, Haiman, Loehr, Remmel and Ulyanov.  ...  Tensoring by the sign representation corresponds to the involution Ω on the space of symmetric functions defined by Ω(s λ ) = s λ ′ , where λ ′ is obtained from λ by transposition.  ...

### Generalized Dyck tilings (Extended Abstract)

Matthieu Josuat-Vergès, Jang Soo Kim
2014 Discrete Mathematics & Theoretical Computer Science
We enumerate symmetric Dyck tilings and show that certain families of symmetric Dyck tilings are in bijection with intervals in the weak order on signed permutations.  ...  Recently, Kenyon and Wilson introduced Dyck tilings, which are certain tilings of the region between two Dyck paths.  ...  Let Dyck (k) (n) denote the set of k-Dyck paths from (0, 0) to (kn, n). A k-Dyck tile is a ribbon in which the centers of the cell form a k-Dyck path.  ...

### Trimer covers in the triangular grid: twenty mostly open problems [article]

James Propp
2022 arXiv   pre-print
Physicists have studied the asymptotics of the trimer model on the triangular lattice but have not given exact results for finite subgraphs of the triangular lattice analogous to known solutions of the  ...  Here I introduce finite subgraphs of the triangular lattice that will be of interest to enumerative combinatorialists inasmuch as the precise number of trimer covers appears to be given by exact formulas  ...  of ribbon tilings [Shef] .  ...

### Generalized Dyck tilings

Matthieu Josuat-Vergès, Jang Soo Kim
2016 European journal of combinatorics (Print)
We enumerate symmetric Dyck tilings and show that certain families of symmetric Dyck tilings are in bijection with intervals in the weak order on signed permutations. Résumé.  ...  Recently, Kenyon and Wilson introduced Dyck tilings, which are certain tilings of the region between two Dyck paths.  ...  Let Dyck (k) (n) denote the set of k-Dyck paths from (0, 0) to (kn, n). A k-Dyck tile is a ribbon in which the centers of the cell form a k-Dyck path.  ...

### Quadrupole moments, edge polarizations, and corner charges in the Wannier representation [article]

Shang Ren, Ivo Souza, David Vanderbilt
2020 arXiv   pre-print
We develop a Wannier-based formalism that allows the corner charge to be predicted, modulo e, only from calculations on ribbon geometries of two different orientations.  ...  The modern theory of polarization allows for the determination of the macroscopic end charge of a truncated one-dimensional insulator, modulo the charge quantum e, from a knowledge of bulk properties alone  ...  The values of P T x , P R y , and Q I xy are obtained from ribbon calculations, while Q TR is inferred mod e from the tiling procedure.  ...

### Adelson's tile and snake illusions: A Helmholtzian type of simultaneous lightness contrast

Alexander Logvinenko, Deborah Ross
2005 Spatial Vision
On the other hand, Helmholtz's idea, that simultaneous lightness contrast originates from an error in judgement of apparent illumination, has been elaborated so as to account for the tile and snake illusions  ...  Adelson's tile, snake, and some other lightness illusions of the same type were measured with the Munsell neutral scale for twenty observers.  ...  as follows from the shadow/lightness invariance) or not.  ...
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