Tetromino tilings and the Tutte polynomial

Some Polyomino Tilings of the Plane

We calculate the generating functions for the number of tilings of rectangles of various widths by the right tromino, the $L$ tetromino, and the $T$ tetromino. This allows us to place lower bounds on the entropy of tilings of the plane by each of these. For the $T$ tetromino, we also derive a lower bound from the solution of the Ising model in two dimensions.

算术级数中的T型四格骨牌（T-tetrominos）

T-Tetrominos in Arithmetic Progression

本文针对T-tetrominos铺砌矩形的经典结论，研究所有合法铺砌中必然存在的有序性程度，以骨牌算术级数的长度作为有序性度量标准。

The tilings of deficient squares by ribbon L-tetrominoes are diagonally cracked

We consider tilings of deficient rectangles by the set $\mathcal{T}_4$ of ribbon $L$-tetrominoes. A tiling exists iff the rectangle is a square of odd side. The missing cell is on the main NW--SE diagonal, in an odd position if the square is $(4m+1)\times (4m+1)$ and in an even position for $(4m+3)\times (4m+3)$. The majority of the tiles in a tiling are paired and each pair tiles a $2\times 4$ rectangle. The tiles in an irregular position and the missing cell form a NW--SE diagonal crack, located in a thin region symmetric about the diagonal, made out of $3\times 3$ squares that overlap over …

用多联骨牌自动枚举铺砌方式

Automatic Enumeration of Tilings by Polyominoes

该研究基于Zeilberger的计算框架，复现L-四格骨牌铺砌的生成函数，还得到宽度5-9的显式生成函数及OEIS未收录的新序列，拓展了多联骨牌铺砌的枚举研究。

The Gap Number of the T-Tetromino

A famous result of D. Walkup states that the only rectangles that may be tiled by the T-tetromino are those in which both sides are a multiple of four. In this paper we examine the rest of the rectangles, asking how many T-tetrominos may be placed into those rectangles without overlap, or, equivalently, what is the least number of gaps that need to be present. We introduce a new technique for exploring such tilings, enabling us to answer this question for all rectangles, up to a small additive constant. We also show that there is some number G such that if both sides of the rectangle are at le…

Hexagonal Tilings: Tutte Uniqueness

We develop the necessary machinery in order to prove that hexagonal tilings are uniquely determined by their Tutte polynomial, showing as an example how to apply this technique to the toroidal hexagonal tiling.