Related papers: The wreath matrix

The McCarty Conjecture states that any McCarty Matrix (an $n\times n$ matrix $A$ with positive integer entries and each of the $2n$ row and column sums equal to $n$), can be additively decomposed into two other matrices, $B$ and $C$, such…

The recursively-constructed family of Mandelbrot matrices $M_n$ for $n=1$, $2$, $\ldots$ have nonnegative entries (indeed just $0$ and $1$, so each $M_n$ can be called a binary matrix) and have eigenvalues whose negatives $-\lambda = c$…

Let $m,n>1$ be integers and $\mathbb{P}_{n,m}$ be the point set of the projective $(n-1)$-space (defined by [2]) over the ring $\mathbb{Z}_m$of integers modulo $m$. Let $A_{n,m}=(a_{uv})$ be the matrix with rows and columns being labeled by…

A classical theorem of Baranyai states that, given integers $2\leq k < n$ such that $k$ divides $n$, one can find a family of ${n-1\choose k-1}$ partitions of $[n]$ into $k$-element subsets such that every subset appears in exactly one…

For integers $n,k,s$, we give a formula for the number $T(n,k,s)$ of order $k$ subsets of the ring $\mathbb{Z}/n\mathbb{Z}$ whose sum of elements is $s$ modulo $n$. To do so, we describe explicitly a sequence of matrices $M(k)$, for…

In 1991, Kahn made the following conjecture. For any $n$-dimensional vector space $V$ and any $n\times n$ array of $n^2$ bases of $V$, it is possible to choose a representative vector from each of these bases in such a way that the…

We show that the wreath Macdonald polynomials for $\mathbb{Z}/\ell\mathbb{Z}\wr\Sigma_n$, when naturally viewed as elements in the vertex representation of the quantum toroidal algebra…

Arrangement graphs were introduced for their connection to computational networks and have since generated considerable interest in the literature. In a pair of recent articles by Chen, Ghorbani and Wong, the eigenvalues for the adjacency…

Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate…

Wreath Macdonald polynomials arise from the geometry of $\Gamma$-fixed loci of Hilbert schemes of points in the plane, where $\Gamma$ is a finite cyclic group of order $r\ge 1$. For $r=1$, they recover the classical (modified) Macdonald…

Consider a crystallographic root system together with its Weyl group $W$ acting on the weight lattice $M$. Let $Z[M]^W$ and $S^*(M)^W$ be the $W$-invariant subrings of the integral group ring $Z[M]$ and the symmetric algebra $S^*(M)$…

The Union Closed Sets Conjecture is one of the most renowned problems in combinatorics. Its appeal lies in the simplicity of its statement contrasted with the potential complexity of its resolution. The conjecture posits that, in any union…

Let $\iota_{k}(m,l)$ denote the total number of intervals of length $m$ across all Dyck paths of semilength $k$ such that each interval contains precisely $l$ falls. We give the formula for $\iota_{k}(m,l)$ and show that…

We construct ensembles of random integrable matrices with any prescribed number of nontrivial integrals and formulate integrable matrix theory (IMT) -- a counterpart of random matrix theory (RMT) for quantum integrable models. A type-M…

Denote by $M_n(K)$ the algebra of $n$ by $n$ matrices with entries in the field $K$. A theorem of Albert and Muckenhoupt states that every trace zero matrix of $M_n(K)$ can be expressed as $AB-BA$ for some pair $(A,B)$ of matrices of…

Let $W_{k,n}^{i}(m)$ denote a matrix with rows and columns indexed by the $k$-subsets and $n$-subsets, respectively, of an $m$-element set. The row $S$, column $T$ entry of $W_{k,n}^{i}(m)$ is $1$ if $|S \cap T| = i$, and is $0$ otherwise.…

We consider wreath product decompositions for semigroups of triangular matrices. We exhibit an explicit wreath product decomposition for the semigroup of all n-by-n upper triangular matrices over a given field k, in terms of aperiodic…

More than twenty-five years ago, Manickam, Mikl\'{o}s, and Singhi conjectured that for positive integers $n,k$ with $n \geq 4k$, every set of $n$ real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ $k$-element subsets whose…

We give a modular branching rule for certain wreath products as a generalization of Kleshchev's modular branching rule for the symmetric groups. Our result contains a modular branching rule for the complex reflection groups $G(m,1,n)$…

We introduce ring theoretic constructions that are similar to the construction of wreath product of groups. In particular, for a given graph $\Gamma=(V,E)$ and an associate algebra $A,$ we construct an algebra $B=A\, wr\, L(\Gamma)$ with…