scholarly journals Enumeration of Hybrid Domino-Lozenge Tilings II: Quasi-Octagonal Regions

10.37236/4669 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Tri Lai
Keyword(s):  
Product Formula ◽  
Cambridge University ◽  
Replacement Method ◽  
Lozenge Tilings ◽  
Simple Product ◽  
Geometric Combinatorics

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.

scholarly journals Lozenge Tilings of Hexagons with Central Holes and Dents

10.37236/8716 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Tri Lai
Keyword(s):  
Product Formula ◽  
Classical Formula ◽  
Plane Partitions ◽  
Equilateral Triangles ◽  
Lozenge Tilings ◽  
Simple Product ◽  
A Chain ◽  
Two Sides

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.  

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

10.37236/9363 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Daniel Condon
Keyword(s):  
Triangular Lattice ◽  
Lozenge Tilings ◽  
Simple Product ◽  
Two Sides ◽  
Product Formulas

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.

scholarly journals A Simple Product Formula for Certain Kazhdan-LusztigR-Polynomials

2006 ◽  
Vol 34 (2) ◽  
pp. 407-414 ◽  
Author(s):  
Michela Pagliacci
Keyword(s):  
Product Formula ◽  
Simple Product

scholarly journals New Aspects of Regions whose Tilings are Enumerated by Perfect Powers

10.37236/3186 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Tri Lai
Keyword(s):  
Triangular Lattice ◽  
Product Formula ◽  
Square Lattice ◽  
Reduction Theorem ◽  
Simple Product ◽  
By Products

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.

scholarly journals The Number of Optimal Matchings for Euclidean Assignment on the Line

2021 ◽  
Vol 183 (1) ◽  
Author(s):  
Sergio Caracciolo ◽  
Vittorio Erba ◽  
Andrea Sportiello
Keyword(s):  
Generating Functions ◽  
Arbitrary Order ◽  
Random Variable ◽  
Singularity Analysis ◽  
Product Formula ◽  
Zero Temperature ◽  
Brownian Process ◽  
The Mean ◽  
Simple Product ◽  
Linear Cost

AbstractWe consider the Random Euclidean Assignment Problem in dimension $$d=1$$ d = 1 , with linear cost function. In this version of the problem, in general, there is a large degeneracy of the ground state, i.e. there are many different optimal matchings (say, $$\sim \exp (S_N)$$ ∼ exp ( S N ) at size N). We characterize all possible optimal matchings of a given instance of the problem, and we give a simple product formula for their number. Then, we study the probability distribution of $$S_N$$ S N (the zero-temperature entropy of the model), in the uniform random ensemble. We find that, for large N, $$S_N \sim \frac{1}{2} N \log N + N s + {\mathcal {O}}\left( \log N \right) $$ S N ∼ 1 2 N log N + N s + O log N , where s is a random variable whose distribution p(s) does not depend on N. We give expressions for the moments of p(s), both from a formulation as a Brownian process, and via singularity analysis of the generating functions associated to $$S_N$$ S N . The latter approach provides a combinatorial framework that allows to compute an asymptotic expansion to arbitrary order in 1/N for the mean and the variance of $$S_N$$ S N .

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