Related papers: Cartesian Magicness of 3-Dimensional Boards
This paper addresses a theory of R(p,q)-deformed combinatorics in discrete probability. It mainly focuses on R(p,q)-deformed factorials, binomial coefficients, Vandermonde's formula, Cauchy's formula, binomial and negative binomial…
We find by applying MacMahon's partition analysis that all magic squares of order three, up to rotations and reflections, are of two types, each generated by three basis elements. A combinatorial proof of this fact is given.
We introduce a method for constructing larger families of connected cospectral graphs from two given cospectral families of sizes $p$ and $q$. The resulting family size depends on the Cartesian primality of the input graphs and can be one…
Let 1_k 0_l denote the (k+l)\times 1 column of k 1's above l 0's. Let q. (1_k 0_l) $ denote the (k+l)xq matrix with q copies of the column 1_k0_l. A 2-design S_{\lambda}(2,3,v) can be defined as a vx(\lambda/3)\binom{v}{2} (0,1)-matrix with…
Several articles deal with tilings with squares and dominoes on 2-dimensional boards, but only a few on boards in 3-dimensional space. We examine a tiling problem with colored cubes and bricks of $(2\times2\times n)$-board in three…
Let $(\Gamma,+)$ be an Abelian group of order $n^2$. A $\Gamma$-magic square of order $n$ is an $n\times n$ array whose entries are pairwise distinct elements of $\Gamma$ such that all row sums, column sums, and the two main diagonal sums…
A triangular partition is a partition whose Ferrers diagram can be separated from its complement (as a subset of $\mathbb{N}^2$) by a straight line. Having their origins in combinatorial number theory and computer vision, triangular…
Four classes of three dimensional quadratic algebras of the type $\lsb Q_0 , Q_\pm \rsb$ $=$ $\pm Q_\pm$, $\lsb Q_+ , Q_- \rsb$ $=$ $aQ_0^2 + bQ_0 + c$, where $(a,b,c)$ are constants or central elements of the algebra, are constructed using…
Magic squares have been an enthralling topic in mathematics for centuries. They are formed by filling in all the cells of a square matrix with the numbers starting from one so that the sum of all rows, columns, and diagonals is the same.…
A triangle group is denoted by $\Delta(p,q,r)$ and has finite presentation $$ \Delta(p,q,r)=\langle x,y | x^p=y^q=(xy)^r=1 \rangle .$$ We examine a method for composition of permutation representations of a triangle group $\Delta(p,q,r)$…
We show arithmetic triplets of Gaussian squares are in 3-to-1 correspondence with Pythagorean triples thereof. This correspondence would transform a solution to the Magic Square of Squares puzzle into a larger structure of perfect Gaussian…
The Stirling numbers of type $B$ of the second kind count signed set partitions. In this paper we provide new combinatorial and analytical identities regarding these numbers as well as Broder's $r$-version of these numbers. Among these…
A number $N$ is a triangular number if it can be written as $N = t(t + 1)/2$ for some nonnegative integer number $t$. A triangular number $N$ is called square if it is a perfect square, that is, $N = d^2$ for some integer number $d$. Square…
A three-dimensional polynomial algebra of order $m$ is defined by the commutation relations $[P_0, P_\pm]$ $=$ $\pm P_\pm$, $[P_+, P_-]$ $=$ $\phi^{(m)}(P_0)$ where $\phi^{(m)}(P_0)$ is an $m$-th order polynomial in $P_0$ with the…
A book of size q is the union of q triangles sharing a common edge. We find the exact Ramsey number of books of size q versus books of size p when p<q/6-o(q).
A {\it cluster of cycles} (or {\it $(r,q)$-polycycle}) is a simple planar 2--co nnected finite or countable graph $G$ of girth $r$ and maximal vertex-degree $q$, which admits {\it $(r,q)$-polycyclic realization} on the plane, denote it by…
A magic labelling of a set system is a labelling of its points by distinct positive integers so that every set of the system has the same sum, the magic sum. Examples are magic squares (the sets are the rows, columns, and diagonals) and…
We consider the tiling of an $n$-board (a board of size $n\times1$) with squares of unit width and $(1,1)$-fence tiles. A $(1,1)$-fence tile is composed of two unit-width square subtiles separated by a gap of unit width. We show that the…
Let $p$ be an odd prime. For nontrivial proper subsets $A,B$ of $\mathbb{Z}_p$ of cardinality $s,t$, respectively, we count the number $r(A,B,B)$ of additive triples, namely elements of the form $(a, b, a+b)$ in $A \times B \times B$. For…
In the theory of Clebsch-Gordan coefficients, one may recognize the domain space as the set of weakly semi-magic squares of size three. Two partitions on this set are considered: a triangle-hexagon model based on top lines, and one based on…