Related papers: Some determinants of path generating functions
We exhibit explicit expressions, in terms of components, of discriminants, determinants, characteristic polynomials and polynomial identities for matrices of higher rank. We define permutation tensors and in term of them we construct…
We show the density of eigenvalues for three classes of random matrix ensembles is determinantal. First we derive the density of eigenvalues of product of $k$ independent $n\times n$ matrices with i.i.d. complex Gaussian entries with a few…
Multiple analogues of certain families of combinatorial numbers are recently constructed by the author in terms of well poised Macdonald functions, and some of their fundamental properties are developed. In this paper, we present…
Recently there has been several works estimating the number of $n\times n$ matrices with elements from some finite sets $\mathcal X$ of arithmetic interest and of a given determinant. Typically such results are compared with the trivial…
Matrix determinants play an important role in data analysis, in particular when Gaussian processes are involved. Due to currently exploding data volumes, linear operations - matrices - acting on the data are often not accessible directly…
This paper presents a method for expressing the determinant of an N {\times} N complex block matrix in terms of its constituent blocks. The result allows one to reduce the determinant of a matrix with N^2 blocks to the product of the…
We summarize some of the recent developments which link certain problems in combinatorial theory related to random growth to random matrix theory.
We prove an interesting identity for the sum of determinants, which is a generalization of the sum of a geometric progression. The proof is quite long and a number of other identities are proved along the way. Some of the more elementary…
Motzkin excursions and meanders are revisited. This is considered in the context of forbidden patterns. Previous work by Asinowski, Banderier, Gittenberger, and Roitner is continued. Motzkin paths of bounded height are considered, leading…
This contribution summarizes recent work of the authors that combines methods from dynamical systems theory (discrete Painlev\'e equations) and asymptotic analysis of orthogonal polynomial recurrences, to address long-standing questions in…
Koutschan, Krattenthaler and Schlosser recently considered a family of binomial determinants. In this work, we give combinatorial interpretations of two subclasses of these determinants in terms of domino tilings and nonintersecting lattice…
We provide formulas for generating functions of many types of paths in various rooted tree structures. We compute the $k$th moment of the generating functions for various types of vertical paths. In two specific familes of trees we find…
A class of parametric functions formed by alternating compositions of multivariate polynomials and rectification style monomial maps is studied (the layer-wise exponents are treated as fixed hyperparameters and are not optimized). For this…
A class of determinants is introduced. Different kind of mathematical objects, such as Fibonacci, Lucas, Tchebychev, Hermite, Laguerre, Legendre polynomials, sums and covergents are represented as determinants from this class. A closed…
We consider two families of random matrix-valued analytic functions: (1) G_1-zG_2 and (2) G_0 + zG_1 +z^2G_2+ ..., where G_i are n x n independent random matrices with independent standard complex Gaussian entries. The set of z where these…
Catalan numbers and their interpretations in terms of Dyck paths are widely used in different topics of applied mathematics and computer science. Here, we consider a general approach for constrained Dyck paths. In particular, we study Dyck…
We present determinantal representations of the Catalan numbers, k-Fuss-Catalan numbers, and its generalized number. The entries of the normalized Hessenberg matrices are the binomial coefficients that related with the enumeration of…
Consideration of a question of E. R. Berlekamp led Carlitz, Roselle, and Scoville to give a combinatorial interpretation of the entries of certain matrices of determinant~1 in terms of lattice paths. Here we generalize this result by…
We survey recent results on determinantal processes, random growth, random tilings and their relation to random matrix theory.
We describe algorithms for computing maximal determinants of binary circulant matrices of small orders. Here "binary matrix" means a matrix whose elements are drawn from $\{0,1\}$ or $\{-1,1\}$. We describe efficient parallel algorithms for…