Related papers: Proof of Blum's conjecture on hexagonal dungeons
Matt Blum conjectured that the number of tilings of a hexagonal dungeon with side-lengths $a,2a,b,a,2a,b$ (for $b\geq2a$) equals $13^{2a^2}14^{\lfloor a^2/2\rfloor}$. Ciucu and the author of the present paper proved the conjecture by using…
We compute the number of rhombus tilings of a hexagon with sides $a+2,b+2,c+2,a+2,b+2,c+2$ with three fixed tiles touching the border. The particular case $a=b=c$ solves a problem posed by Propp. Our result can also be viewed as the…
The combinatorics of tilings of a hexagon of integer side-length $n$ by 120 degree - 60 degree diamonds of side-length 1 has a long history, both directly (as a problem of interest in thermodynamic models) and indirectly (through the…
Conway and Lagarias showed that certain roughly triangular regions in the hexagonal grid cannot be tiled by shapes Thurston later dubbed tribones. Here we study a two-parameter family of roughly hexagonal regions in the hexagonal grid and…
In their 2002 paper, Ciucu and Krattenthaler proved several product formulas for the number of lozenge tilings of various regions obtained from a centrally symmetric hexagon on the triangular lattice by removing maximal staircase regions…
It has been proven that the lozenge tilings of a quartered hexagon on the triangular lattice are enumerated by a simple product formula. In this paper we give a new proof for the tiling formula by using Kuo's graphical condensation. Our…
We compute the number of all rhombus tilings of a hexagon with sides $a,b+1,c,a+1,b,c+1$, of which the central triangle is removed, provided $a,b,c$ have the same parity. The result is a product of four numbers, each of which counts the…
A technique called graphical condensation is used to prove various combinatorial identities among numbers of (perfect) matchings of planar bipartite graphs and tilings of regions. Graphical condensation involves superimposing matchings of a…
In the last decade there have been many results about special families of graphs whose number of perfect matchings is given by perfect or near perfect powers. In this paper we present an approach that allows proving them in a unified way.…
The question of whether a given region can be successfully filled by a finite set of tiles has been commonly studied, and there are many available arguments for whether a given finite region can be tiled. We can show that there is no domino…
We develop the basic tools for classifying edge-to-edge tilings of the sphere by congruent pentagons. Then we prove that, for the edge combination $a^2b^2c$, such tilings are three two-parameter families of pentagonal subdivisions of the…
The classical honeycomb conjecture asserts that any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling. Pappus discusses this problem in his preface to Book V. This paper…
Propp conjectured that the number of lozenge tilings of a semiregular hexagon of sides $2n-1$, $2n-1$ and $2n$ which contain the central unit rhombus is precisely one third of the total number of lozenge tilings. Motivated by this, we…
We develop a recursive formula for counting the number of rectangulations of a square, i.e the number of combinatorially distinct tilings of a square by rectangles. Our formula specializes to give a formula counting generic rectangulations,…
Proctor proved a formula for the number of lozenge tilings of a hexagon with side-lengths $a,b,c,a,b,c$ after removing a "maximal staircase." Ciucu then presented a weighted version of Proctor's result. Here we present weighted and…
We solve and generalize an open problem posted by James Propp (Problem 16 in New Perspectives in Geometric Combinatorics, Cambridge University Press, 1999) on the number of tilings of quasi-hexagonal regions on the square lattice with every…
Rohatgi and the author recently proved a shuffling theorem for lozenge tilings of `doubly-dented hexagons' (arXiv:1905.08311). The theorem can be considered as a hybrid between two classical theorems in the enumeration of tilings:…
The enumeration of lozenge tilings of hexagons with holes has received much attention during the last three decades. One notable feature is that a lot of the recent development involved Kuo's graphical condensation. Motivated by Ciucu, Lai…
In arXiv:1905.08311, the author and Rohatgi proved a shuffling theorem for doubly-dented hexagons. In particular, we showed that shuffling removed unit triangles along a horizontal axis in a hexagon only changes the tiling number by a…
We study the Tur\'{a}n problem for highly symmetric bipartite graphs arising from geometric shapes and periodic tilings commonly found in nature. 1. The prism $C_{2\ell}^{\square}:=C_{2\ell}\square K_{2}$ is the graph consisting of two…