Related papers: Minimal transitive factorizations of permutations …

Factorial designs have broad applications in agricultural, engineering and scientific studies. In constructing and studying properties of factorial designs, traditional design theory treats all factors as nominal. However, this is not…

We introduce the $D$-decomposition, a non-orthogonal matrix factorization of the form $A \approx P D Q$, where $P \in \mathbb{R}^{n \times k}$, $D \in \mathbb{R}^{k \times k}$, and $Q \in \mathbb{R}^{k \times n}$. The decomposition is…

The number of minimal transitive star factorizations of a permutation was shown by Irving and Rattan to depend only on the conjugacy class of the permutation, a surprising result given that the pivot plays a very particular role in such…

At the end of the 1960s, Knuth characterised the permutations that can be sorted using a stack in terms of forbidden patterns. He also showed that they are in bijection with Dyck paths and thus counted by the Catalan numbers. Subsequently,…

The elliptic curve method (ECM) is one of the best factorization methods available. It is possible to use hyperelliptic curves instead of elliptic curves but it is in theory slower. We use special hyperelliptic curves and Kummer surfaces to…

Lifted probabilistic inference exploits symmetries in probabilistic graphical models to allow for tractable probabilistic inference with respect to domain sizes. To exploit symmetries in, e.g., factor graphs, it is crucial to identify…

We use the 0-1 tableaux as a tool for enumerating permutations and partitions with restricted minima. The method used is extended for permutation pairs and partition pairs generated by a bipartite 0-1 tableaux.

Motivated by coding applications,two enumeration problems are considered: the number of distinct divisors of a degree-m polynomial over F = GF(q), and the number of ways a polynomial can be written as a product of two polynomials of degree…

This paper proposes a new perspective on the problem of multidimensional spectral factorization, through helical mapping: $d$-dimensional ($d$D) data arrays are vectorized, processed by $1$D cepstral analysis and then remapped onto the…

We consider the lattice of all the weak factorization systems on a given finite lattice. We prove that it is semidistributive, trim and congruence uniform. We deduce a graph theoretical approach to the problem of enumerating transfer…

The problem of enumerating meanders -- pairs of simple plane curves with transverse intersections -- was formulated about forty years ago and is still far from solved. Recently, it was discovered that meanders admit a factorization into…

Reducing parameter redundancies in neural network architectures is crucial for achieving feasible computational and memory requirements during training and inference phases. Given its easy implementation and flexibility, one promising…

We combine the language of monoids with the language of preorders so as to refine some fundamental aspects of the classical theory of factorization and prove an abstract factorization theorem with a variety of applications. In particular,…

We consider the problem of enumerating d-irreducible maps, i.e. planar maps whose all cycles have length at least d, and such that any cycle of length d is the boundary of a face of degree d. We develop two approaches in parallel: the…

We study the deformation theory aspects of Matricial Factorizations, possibly with an orthogonal or symplectic structure. We discuss and extend the Kn\"orrer and Hori-Walcher periodicity theorems

Billey, Jockusch, and Stanley characterized 321-avoiding permutations by a property of their reduced decompositions. This paper generalizes that result with a detailed study of permutations via their reduced decompositions and the notion of…

As well known, permanent of a square (0,1)-matrix $A$ of order $n$ enumerates the permutations $\beta$ of $1,2,...,n$ with the incidence matrices $B\leq A.$ To obtain enumerative information on even and odd permutations with condition…

We use the periodicity properties of generalized Gauss sums to factor numbers. Moreover, we derive rules for finding the factors and illustrate this factorization scheme for various examples. This algorithm relies solely on interference and…

Despite the popularity of factor models with sparse loading matrices, little attention has been given to formally address identifiability of these models beyond standard rotation-based identification such as the positive lower triangular…

Unique factorization fails in many rings and monoids, but divisor and transfer homomorphisms provide tools to understand non-unique factorizations. In this expository article, we first explore these notions in the classical setting of…