Related papers: Enumerating partial Latin rectangles

We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work of Norbury that, if one disallows vertices of degree one, the enumeration of such maps is related to the counting of lattice…

Let $A$ be a $r\times r$ rational nonzero symmetric matrix, $B$ a rational column vector, $C$ a rational scalar. For any integer lattice $L$ and vector $v$ of $\mathbb{Z}^r$, we define Nahm sum on the lattice coset $v+L\in \mathbb{Z}^r/L$:…

What is the smallest number of pieces that you can cut an n-sided regular polygon into so that the pieces can be rearranged to form a rectangle? Call it r(n). The rectangle may have any proportions you wish, as long as it is a rectangle.…

We study universal polynomials of characteristic classes associated to the $\mathcal{A}$-classification (i.e. up to right-left equivalence) of holomorphic map-germs $(\mathbb{C}^2,0) \to (\mathbb{C}^n, 0)$ $(n=2,3)$. That enables us to…

The purpose of this short article is to announce, and briefly describe, a Maple package, PARTITIONS, that (inter alia) completely automatically discovers, and then proves, explicit expressions (as sums of quasi-polynomials) for pm(n) for…

We consider the problem of exhaustively visiting all pairs of linear cellular automata which give rise to orthogonal Latin squares, i.e., linear Orthogonal Cellular Automata (OCA). The problem is equivalent to enumerating all pairs of…

Young's partition lattice $L(m,n)$ consists of unordered partitions having $m$ parts where each part is at most $n$. Using methods from complex algebraic geometry, R. Stanley proved that $L(m,n)$ is rank-symmetric, unimodal, and strongly…

This article, showing that almost all objects in the title are asymmetric, is re-typed from a manuscript I wrote somewhere around 1980 (after the papers of Bang and Friedland on the permanent conjecture but before those of Egorychev and…

We introduce the \emph{Generalized Latin Square Graph} $\Gamma(S)$ of a finite semigroup $S$. Since we record global factorization multiplicities and local alternative counts, we define three counting invariants $N_S,N_R,N_C$. This gives…

We study tilings of rectangular boards using unit squares together with a single type of big tile shaped as a Ferrers diagram. We derive generating functions for these tilings, prove real-rootedness and interlacing properties of associated…

It is shown that if $F$ denotes the number of filled cells in a superimposed pair of maximal orthogonal partial Latin squares of order $n$, then $F\ge n^2/3$. This resolves a conjecture raised in an earlier paper by the current authors. It…

A $(k; r, s; n, q)$-set (short: $(r,s)$-set) of $\mathrm{PG}(n, q)$ is a set of points $X$ with $|X| = k$ such that no $s$-space contains more than $r$ points of $X$. We investigate the asymptotic size of $(r, s)$-sets for $n$ fixed and $q…

This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…

Let $m\ge 2$ be a fixed positive integer. Suppose that $m^j \leq n< m^{j+1}$ is a positive integer for some $j\ge 0$. Denote $b_{m}(n)$ the number of $m$-ary partitions of $n$, where each part of the partition is a power of $m$. In this…

Let $\sigma_b(X_{m,d}(\mathbb {C}))(\mathbb {R})$, $b(m+1) < \binom{m+d}{m}$, denote the set of all degree $d$ real homogeneous polynomials in $m+1$ variables (i.e. real symmetric tensors of format $(m+1)\times ... \times (m+1)$, $d$ times)…

In this paper we introduce the polynomials $\{d_n^{(r)}(x)\}$ and $\{D_n^{(r)}(x)\}$ given by $d_n^{(r)}(x)=\sum_{k=0}^n\binom{x+r+k}k\binom{x-r}{n-k} \ (n\ge 0)$, $D_0^{(r)}(x)=1,\ D_1^{(r)}(x)=x$ and…

We investigate the coefficients of the polynomial \[ S_{m,r}^n(\ell)=r^n+(m+r)^n+(2m+r)^n+\cdots+((\ell-1)m+r)^n. \] We prove that these can be given in terms of Stirling numbers of the first kind and $r$-Whitney numbers of the second kind.…

An intercalate in a Latin square is a $2\times2$ Latin subsquare. Let $N$ be the number of intercalates in a uniformly random $n\times n$ Latin square. We prove that asymptotically almost surely…

In this paper we obtain the formal asymptotic expansion of the logarithms $\ln p_s(\alpha)$ of $p_s(\alpha)$, which are canonical continuations of polynomials of binomial type $p_n(\alpha)$. Our approach is based on linear methods which do…

Given the collection of all $m\times n$ rectangular grids which have a fixed number $1\leq r\leq mn$ of blocked cells, we explicitly describe a proper subset of the collection which is guaranteed to contain at least one grid from each…