Related papers: Enumerating extensions of mutually orthogonal Lati…

In this paper, we generalize the Catalan number to the $(n,k)$-th Catalan numbers and find a combinatorial description that the $(n,k)$-th Catalan numbers is equal to the number of partitions of $n(k-1)+2$ polygon by $(k+1)$-gon where all…

The one variable Bernstein-Szego theory for orthogonal polynomials on the real line is extended to a class of two variable measures. The polynomials orthonormal in the total degree ordering and the lexicographical ordering are constructed…

This paper deals with distinct computational methods to enumerate the set $\mathrm{PLR}(r,s,n;m)$ of $r \times s$ partial Latin rectangles on $n$ symbols with $m$ non-empty cells. For fixed $r$, $s$, and $n$, we prove that the size of this…

Let $k\geq\ell\geq1$ and $n\geq 1$ be integers. Let $G(k,n)$ be the complete $k$-partite graph with $n$ vertices in each colour class. An $\ell$-decomposition of $G(k,n)$ is a set $X$ of copies of $K_k$ in $G(k,n)$ such that each copy of…

A latin bitrade (T1, T2) is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. A genus may be associated to a latin bitrade…

For $\mu$ given latin squares of order $n$, they have {\sf $k$ intersection} when they have $k$ identical cells and $n^2-k$ cells with mutually different entries. For each $n\geq 1$ the set of integers $k$ such that there exist $\mu$ latin…

After a short introduction to anti-linearity, bounds for the number of orthogonal (skew) conjugations are proved. They are saturated if the dimension of the Hilbert space is a power of two. For the other dimensions this is an open problem.

Let $\{\mathbb{P}_n\}_{n\ge 0}$ and $\{\mathbb{Q}_n\}_{n\ge 0}$ be two monic polynomial systems in several variables satisfying the linear structure relation $$\mathbb{Q}_n = \mathbb{P}_n + M_n \mathbb{P}_{n-1}, \quad n\ge 1,$$ where $M_n$…

A strongly reflective modular form with respect to an orthogonal group of signature (2,n) determines a Lorentzian Kac--Moody algebra. We find a new geometric application of such modular forms: we prove that if the weight is larger than n…

Let P be a polygon whose vertices have been colored (labeled) cyclically with the numbers 1,2,...,c. Motivated by conjectures of Propp, we are led to consider partitions of P into k-gons which are proper in the sense that each k-gon…

Let $n=hw$, where $h$ and $w$ are integers with $h,w \ge 2$. We determine the set of possible intersection numbers of two $n \times n$ latin squares having the additional `Sudoku' constraint based on a $w \times h$ grid of $h \times w$…

A hypergraph is \textit{bipartite with bipartition $(A, B)$} if every edge has exactly one vertex in $A$, and a matching in such a hypergraph is \textit{$A$-perfect} if it saturates every vertex in $A$. We prove an upper bound on the number…

We show that for every $d\ge 1$, if $L_1,\ldots, L_d$ are linearly ordered compact spaces and there is a continuous surjection \[ L_1\times L_2\times \dots\times L_d\to K_1\times K_2\times\ldots\times K_{d}\times K_{d+1},\] where all the…

Aougab and Gaster [Math. Proc. Cambridge Philos. Soc. 174 (2023), 569-584] proved that any set of simple closed curves on the torus, where any two are non-homotopic and intersect at most k times, has a maximum size of $k+O(\sqrt{k}\log k)$.…

In a convex n-gon, let d[1] > d[2] > ... denote the set of all distances between pairs of vertices, and let m[i] be the number of pairs of vertices at distance d[i] from one another. Erdos, Lovasz, and Vesztergombi conjectured that m[1] +…

Following the earlier work on {homogeneous Latin bitrades by Cavenagh, Donovan, and Dr'apal (2003 and 2004) Bean, Bidkhori, Khosravi, and E. S. Mahmoodian (2005) we prove the following results. All k-homogeneous Latin bitrades of volume km…

This is a little investigation into the classification of complexes of direct sums of line bundles on projective spaces. We consider complexes on projective k-space Pk : O_Pk(-1)^a --> O_Pk^b --> O_Pk(1)^c, with the first map injective and…

There exist pairs of orthogonal Latin squares of any order n except if n=2 or n=6 [Bose, Shrikhande and Parker, 1960]. In particular, the problem of Euler's thirty-six officers does not have a solution. However, it has a "quantum solution":…

We study the number of linear extensions of a partial order with a given proportion of comparable pairs of elements, and estimate the maximum and minimum possible numbers. We also consider a random interval partial order on $n$ elements,…

The moduli space of triangles is a two-dimensional space that records triangle shapes in the plane, considered up to similarity. We study the subset corresponding to \textit{lattice triangles}, which are triangles whose vertices have…