Related papers: From geometry to generating functions: rectangulat…
Main purpose of this paper is to reconstruct generating function of the Bernstein type polynomials. Some properties this generating functions are given. By applying this generating function, not only derivative of these polynomials but also…
We argue that the mathematical structure, enabling certain cascading and emergent phenomena to intuitively emerge, coincides with Galois connections. We introduce the notion of generative effects to formally capture such phenomena. We…
We modify the enumeration schemes of Zeilberger and Vatter so that they can efficiently enumerate many new classes of pattern-avoiding permutations including all such classes with a regular insertion encoding.
We develop the theory of fractal homeomorphisms generated from pairs of overlapping affine iterated function systems.
We explore the question concerning the number of distinct resonant algebras depending on the generator content, which consists of the Lorentz generator, translation, and new additional Lorentz-like and translation-like generators. Such…
We study aspects of the enumeration of permutation classes, sets of permutations closed downwards under the subpermutation order. First, we consider monotone grid classes of permutations. We present procedures for calculating the generating…
We characterize bijections on matrix spaces (operator algebras) preserving full rank (invertibility) of differences of matrix (operator) pairs in both directions.
Let $c_n$ denote the number of nodes at a distance $n$ from the root of a rooted tree. A criterion for proving the rationality and computing the rational generating function of the sequence $\{c_n\}$ is described. This criterion is applied…
Ouroboros functions have shown some interesting properties when subjected to conventional operations. The aim of this paper is to continue our investigation and prove some additional properties of these functions. Using algebraic methods,…
In this paper we continue the study of permutations avoiding the vincular pattern $1-32-4$ by constructing a generating tree with a single label for these permutations. This construction finally provides a clearer explanation of why a…
Working with generating functions, the combinatorics of a recurrence relation can be expressed in a way that allows for more efficient calculation of the quantity. This is true of the Catalan numbers for an ordered binary tree…
Understanding network functionality requires integrating structure and dynamics, and emergent latent geometry induced by network-driven processes captures the low-dimensional spaces governing this interplay. Here, we focus on…
Here we consider the degenerate Bernstein polynomials as a degenerate version of Bernstein polynomials, which are motivated by Simsek's recent work 'Generating functions for unification of the multidimensional Bernstein polynomials and…
Factoring out the spin $1$ subalgebra of a $ W $ algebra leads to a new $ W $ structure which can be seen either as a rational finitely generated $ W $ algebra or as a polynomial non-linear $ W_\infty$ realization.
Our main theorem classifies the Auslander-Reiten triangles according to properties of the morphisms involved. As a consequence, we are able to compute the mapping cone of an irreducible morphism. We finish by showing a technique for…
Certain classical generating functions for elements of reflection groups can be expressed using fundamental invariants called exponents. We give new analogues of such generating functions that accommodate orbits of reflecting hyperplanes…
We consider the set of affine permutations that avoid a fixed permutation pattern. Crites has given a simple characterization for when this set is infinite. We find the generating series for this set using the Coxeter length statistic and…
This dissertation presents a multifaceted look into the structural decomposition of permutation classes. The theory of permutation patterns is a rich and varied field, and is a prime example of how an accessible and intuitive definition…
We describe generating functions for several important families of classical symmetric functions and shifted Schur functions. The approach is originated from vertex operator realization of symmetric functions and offers a unified method to…
We define a class L_{n, k} of permutations that generalizes alternating (up-down) permutations and give bijective proofs of certain pattern-avoidance results for this class. As a special case of our results, we give two bijections between…