EQUIVALENCE CLASSES AMONG PENTOMINO TILINGS OF THE 6xl0 RECTANGLE

Counting Fixed-Height Tatami Tilings

A tatami tiling is an arrangement of 1 × 2 dominoes (or mats) in a rectangle with m rows and n columns, subject to the constraint that no four corners meet at a point. For fixed m we present and use Dean Hickerson’s combinatorial decomposition of the set of tatami tilings — a decomposition that allows them to be viewed as certain classes of restricted compositions when n ≥ m. Using this decompo...

The graph of equivalence classes and Isoclinism of groups

‎Let $G$ be a non-abelian group and let $Gamma(G)$ be the non-commuting graph of $G$‎. ‎In this paper we define an equivalence relation $sim$ on the set of $V(Gamma(G))=Gsetminus Z(G)$ by taking $xsim y$ if and only if $N(x)=N(y)$‎, ‎where $ N(x)={uin G | x textrm{ and } u textrm{ are adjacent in }Gamma(G)}$ is the open neighborhood of $x$ in $Gamma(G)$‎. ‎We introduce a new graph determined ...

On Counting the Number of Tilings of a Rectangle with Squares of Size 1 and 2

We consider tilings of a rectangle of size n×k with square tiles of size 1×1 and 2×2. We present a method to calculate the number of such tilings via matrix multiplication, where we optimize the number of multiplications needed and reduce the space required for the matrix multiplication by dynamically generating the matrices involved.

Rhombic Tilings of Polygons and Classes of Reduced Words in Coxeter Groups

In the standard Coxeter presentation, the symmetric group Sn is generated by the ad jacent transpositions (1,2), (2,3), ... , (n1,n). For any given permutation, we consider all minimal-length factorizations thereof as a product of the generators. Any two transpositions (i,i+1) and (j,j+1) commute if the numbers i and j are not consecutive; thus, in any factorization, their order can be switche...

Some equivalence classes of operators on B(H)