Related papers: Inverting the General Order Sweep Map
This work presents a generalized notion of multiset mapping thus resolving a long standing obstacle in structural study of multiset processing. It has been shown that the mapping defined herein can model a vast array of notions as special…
Given a finite-dimensional, complex simple Lie algebra we exhibit an integral form for the universal enveloping algebra of its map algebra, and an explicit integral basis for this integral form. We also produce explicit commutation formulas…
In his recent research M. K. Tam (2018) considered a framework for the analysis of iterative algorithms which can be described in terms of a structured set-valued operator. At each point in the ambient space, the value of the operator can…
We present a generalization, which we call (k,m)-rank, of Dyson's notion of rank to integer partitions with k successive Durfee rectangles and give two combinatorial symmetries associated with this new definition. We prove these symmetries…
We give bijective results between several variants of lattice paths of length $2n$ (or $2n-2$) and integer compositions of n, all enumerated by the seemingly innocuous formula $4^{n-1}$. These associations lead us to make new connections…
Combinatorial spiders are a model for the invariant space of the tensor product of representations. The basic objects, webs, are certain directed planar graphs with boundary; algebraic operations on representations correspond to…
A {\em k-generalized Dyck path} of length $n$ is a lattice path from $(0,0)$ to $(n,0)$ in the plane integer lattice $\mathbb{Z}\times\mathbb{Z}$ consisting of horizontal-steps $(k, 0)$ for a given integer $k\geq 0$, up-steps $(1,1)$, and…
In this paper, we introduce two new forms of the dual Hartwig-Spindelb{\"o}ck decomposition and employ them to derive explicit representations for several classes of dual generalized inverses. Building on these representations, we further…
We give a bijective proof of a conjecture of Regev and Vershik on the equality of two multisets of hook numbers of certain skew-Young diagrams. The bijection proves a result that is stronger and more symmetric than the original conjecture,…
A recurrence relation of the generating function of the dimer model of Fibonacci type gives a functional relation for formal power series associated to lattice paths such as a Dyck, Motzkin and Schr\"oder path. In this paper, we generalize…
In the present paper authors introduce the L_n-integral transform and the inverse integral transform for n = 2^k, k=0,1,2,..., as a generalization of the classical Laplace transform and the inverse Laplace transform, respectively.…
We unify and extend previous bijections on plane quadrangulations to bipartite and quasibipartite plane maps. Starting from a bipartite plane map with a distinguished edge and two distinguished corners (in the same face or in two different…
Recent advances in AI have been primarily driven by large-scale neural architectures that excel at function approximation, rather than by tailored inductive biases and inference or learning strategies that could be important for…
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 consider the family of generalized Paley graphs (GP-graphs for short) $\Gamma(k,q) = Cay(\mathbb{F}_q, (\mathbb{F}_q^*)^k)$, with $q=p^m$ and $p$ prime. We characterize all GP-graphs having real spectrum; namely, $Spec(\Gamma(k,q))…
We develop a theory of umkehr maps for twisted generalized homology theories. In this theory, interesting umkehr maps, including generalizations of important classical ones, are induced by cartesian morphisms of a certain category opfibred…
We consider the set of alternating paths on a fixed fully packed loop of size n. This set is in bijection with the set of fully packed loops of size n. Furthermore, for a special choice of fully packed loop, we demonstrate that the set of…
In this note we give an algorithm to determine the rational homotopy type of the free and pointed mapping spaces $ map(F(\mathbb R^m,k), S^n)$ and $ map^*(F(\mathbb R^m,k), S^n)$. An explicit description of these spaces is given for $k=3$.…
It is well-known and easy to show that even the following version of the directed travelling salesman problem is NP-complete: Given a strongly connected complete digraph $D=(V,A)$, a cost function $w: A\rightarrow \{0,1\}$ and a natural…
To approximate a simple root of an equation we construct families of iterative maps of higher order of convergence. These maps are based on model functions which can be written as an inner product. The main family of maps discussed is…