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The least square solution of minimum norm of a rectangular linear system of equations can be found out iteratively by using matrix splittings. However, the convergence of such an iteration scheme arising out of a matrix splitting is…
The theme of this article is a "reciprocity" between bounded up-down paths and bounded alternating sequences. Roughly speaking, this ``reciprocity" manifests itself by the fact that the extension of the sequence of numbers of paths of…
This paper studies the problem of decomposing a low-rank matrix into a factor with binary entries, either from $\{\pm 1\}$ or from $\{0,1\}$, and an unconstrained factor. The research answers fundamental questions about the existence and…
Integer partitions are one of the most fundamental objects of combinatorics (and number theory), and so is enumerating objects avoiding patterns. In the present paper we describe two approaches for the systematic counting of classes of…
After a short review of the Method of Recursive Counting we introduce a general algebraic description of recursive lattice building. This provides a rigorous framework for discussion of method's limitations.
Several articles deal with tilings with various shapes, and also a very frequent type of combinatorics is to examine the walks on graphs or on grids. We combine these two things and give the numbers of the shortest walks crossing the tiled…
Matrices can be augmented by adding additional columns such that a partitioning of the matrix in blocks of rows defines mutually orthogonal subspaces. This augmented system can then be solved efficiently by a sum of projections onto these…
The determination of the density functions for products of random elements from specified classes of matrices is a basic problem in random matrix theory and is also of interest in theoretical physics. For connected simple Lie groups of…
We consider the problem of enumerating hypermatrices of format $2 \times (k + 1) \times k$ over a finite field that have nonzero hyperdeterminant and whose nonzero entries are restricted to a plane partition. We conjecture an attractive…
The weak saturation number $\mathrm{wsat}(n,F)$ is the minimum number of edges in a graph on $n$ vertices such that all the missing edges can be activated sequentially so that each new edge creates a copy of $F$. A usual approach to prove a…
We propose a new method to enumerate alternating knots using a transfer matrix approach. We apply it to count numerically various objects, including prime alternating tangles with two connected components, up to order 18--22, and comment on…
In this paper, we count the number of matrices whose rows generate different $\mathbb{Z}_2\mathbb{Z}_8$ additive codes. This is a natural generalization of the well known Gaussian numbers that count the number of matrices whose rows…
Recombining trinomial trees are a workhorse for modeling discrete-event systems in option pricing, logistics, and feedback control. Because each node stores a state-dependent quantity, a depth-$D$ tree naively yields $\mathcal{O}(3^{D})$…
An algorithm for obtaining all n\times n binary matrices having exactly 2 units in every row and every column is described in the paper. After analysing the work of the algorithm a formula for calculating the number of these matrices has…
We introduce weak exceptional sequence of modules which can be viewed as another modification of the standard case, different than the works of Igusa-Todorov \cite{Igusa-Todorov} and Buan-Marsh \cite{Buan-Marsh}. For hereditary algebras it…
We report on some recent progress made in understanding weak matrix elements of mesons in the context of the next-to-leading order of the large-Nc approximation to QCD. Specifically, we first use the example of the weak contributions to the…
This paper presents a bijection between ascent sequences and upper triangular matrices whose non-negative entries are such that all rows and columns contain at least one non-zero entry. We show the equivalence of several natural statistics…
We count the number of alignments of $N \ge 1$ sequences when match-up types are from a specified set $S\subseteq \mathbb{N}^N$. Equivalently, we count the number of nonnegative integer matrices whose rows sum to a given fixed vector and…
For an odd integer $r>0$ and an integer $n>r$, we introduce a notion of weakly $r$-separated collections of subsets of $[n]=\{1,2,\ldots,n\}$. When $r=1$, this corresponds to the concept of weak separation introduced by Leclerc and…
The set of permutations on a finite set can be given the lattice structure known as the weak Bruhat order. This lattice structure is generalized to the set of words on a fixed alphabet $\Sigma$ = {x,y,z,...}, where each letter has a fixed…