scholarly journals Lozenge Tiling Function Ratios for Hexagons with Dents on Two Sides

10.37236/9363 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Daniel Condon

We give a formula for the number of lozenge tilings of a hexagon on the triangular lattice with unit triangles removed from arbitrary positions along two non-adjacent, non-opposite sides. Our formula implies that for certain families of such regions, the ratios of their numbers of tilings are given by simple product formulas.

10.37236/8716 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Tri Lai

Ciucu proved a simple product formula for the tiling number of a hexagon in which a chain of equilateral triangles of alternating orientations, called a `fern', has been removed from the center (Adv. Math. 2017). In this paper, we present a multi-parameter generalization of this work by giving an explicit tiling enumeration for a hexagon with three ferns removed, besides the central fern as in Ciucu's region, we remove two new ferns from two sides of the hexagon. Our result also implies a new `dual' of MacMahon's classical formula of boxed plane partitions, corresponding to the exterior of the union of three disjoint concave polygons obtained by turning 120 degrees after drawing each side.  


10.37236/4669 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Tri Lai

We use the subgraph replacement method to prove a simple product formula for the tilings of an  8-vertex counterpart of Propp's quasi-hexagons (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999), called quasi-octagon.


10.37236/3186 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Tri Lai

In 2003, Ciucu presented a unified way to enumerate tilings of lattice regions by using a certain Reduction Theorem (J. Algebraic Combin., 2003). In this paper we continue this line of work by investigating new families of lattice regions whose tilings are enumerated by perfect powers or products of several perfect powers. We prove a multi-parameter generalization of Bo-Yin Yang's theorem on fortresses (Ph.D. thesis, MIT, 1991).  On the square lattice with zigzag paths, we consider two particular families of regions whose numbers of tilings are always a power of 3 or twice a power of 3. The latter result provides a new proof for a conjecture of Matt Blum first proved by Ciucu. We also consider several new lattices obtained by periodically applying two simple subgraph replacement rules to the square lattice. On some of those lattices, we get new families of regions whose numbers of tilings  are given by products of several perfect powers. In addition, we prove a simple product formula for the number of tilings of a certain family of regions on a variant of the triangular lattice.


10.37236/3429 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Tri Lai

We get four quartered Aztec diamonds by dividing an Aztec diamond region by two zigzag cuts passing its center. W. Jockusch and J. Propp (in an unpublished work) found that the number of tilings of quartered Aztec diamonds is given by simple product formulas. In this paper we present a simple proof for this result.


10.37236/7502 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Tri Lai

Proctor's work on staircase plane partitions yields an exact enumeration of lozenge tilings of a halved hexagon on the triangular lattice. Rohatgi later extended this tiling enumeration to a halved hexagon with a triangle cut off from the boundary. In his previous paper, the author proved  a common generalization of Proctor's and Rohatgi's results by enumerating lozenge tilings of a halved hexagon in the case an array of an arbitrary number of triangles has been removed from a non-staircase side. In this paper we consider the other case when the array of triangles has been removed from the staircase side of the halved hexagon. Our result also implies an explicit formula for the number of tilings of a hexagon with an array of triangles removed perpendicularly to the symmetry axis.


Author(s):  
C. Goessens ◽  
D. Schryvers ◽  
J. Van Landuyt ◽  
A. Verbeeck ◽  
R. De Keyzer

Silver halide grains (AgX, X=Cl,Br,I) are commonly recognized as important entities in photographic applications. Depending on the preparation specifications one can grow cubic, octahedral, tabular a.o. morphologies, each with its own physical and chemical characteristics. In the present study crystallographic defects introduced by the mixing of 5-20% iodide in a growing AgBr tabular grain are investigated. X-ray diffractometry reveals the existence of a homogeneous Ag(Br1-xIx) region, expected to be formed around the AgBr kernel. In fig. 1 a two-beam BF image, taken at T≈100 K to diminish radiation damage, of a triangular tabular grain is presented, clearly showing defect contrast fringes along four of the six directions; the remaining two sides show similar contrast under relevant diffraction conditions. The width of the central defect free region corresponds with the pure AgBr kernel grown before the mixing with I. The thickness of a given grain lies between 0.15 and 0.3 μm: as indicated in fig. 2 triangular (resp. hexagonal) grains exhibit an uneven (resp. even) number of twin interfaces (i.e., between + and - twin variants) parallel with the (111) surfaces. The thickness of the grains and the existence of the twin variants was confirmed from CTEM images of perpendicular cuts.


2004 ◽  
Vol 32 (1) ◽  
pp. 181-184
Author(s):  
Amy Garrigues

On September 15, 2003, the US. Court of Appeals for the Eleventh Circuit held that agreements between pharmaceutical and generic companies not to compete are not per se unlawful if these agreements do not expand the existing exclusionary right of a patent. The Valley DrugCo.v.Geneva Pharmaceuticals decision emphasizes that the nature of a patent gives the patent holder exclusive rights, and if an agreement merely confirms that exclusivity, then it is not per se unlawful. With this holding, the appeals court reversed the decision of the trial court, which held that agreements under which competitors are paid to stay out of the market are per se violations of the antitrust laws. An examination of the Valley Drugtrial and appeals court decisions sheds light on the two sides of an emerging legal debate concerning the validity of pay-not-to-compete agreements, and more broadly, on the appropriate balance between the seemingly competing interests of patent and antitrust laws.


Author(s):  
Ángel Correa ◽  
Paola Cappucci ◽  
Anna C. Nobre ◽  
Juan Lupiáñez

Would it be helpful to inform a driver about when a conflicting traffic situation is going to occur? We tested whether temporal orienting of attention could enhance executive control to select among conflicting stimuli and responses. Temporal orienting was induced by presenting explicit cues predicting the most probable interval for target onset, which could be short (400 ms) or long (1,300 ms). Executive control was measured both by flanker and Simon tasks involving conflict between incompatible responses and by the spatial Stroop task involving conflict between perceptual stimulus features. The results showed that temporal orienting facilitated the resolution of perceptual conflict by reducing the spatial Stroop effect, whereas it interfered with the resolution of response conflict by increasing flanker and Simon effects. Such opposite effects suggest that temporal orienting of attention modulates executive control through dissociable mechanisms, depending on whether the competition between conflicting representations is located at perceptual or response levels.


2011 ◽  
Author(s):  
Stephanie A. Quezada ◽  
Isabel J. Gonzalez ◽  
Michael A. Zarate

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