Related papers: Some determinants of path generating functions
We give a combinatorial interpretation of the determinant of a matrix as a generating function over Brauer diagrams in two different but related ways. The sign of a permutation associated to its number of inversions in the Leibniz formula…
We establish three identities involving Dyck paths and alternating Motzkin paths, whose proofs are based on variants of the same bijection. We interpret these identities in terms of closed random walks on the halfline. We explain how these…
For a restricted class of potentials (harmonic+Gaussian potentials), we express the resolvent integral for the correlation functions of simple traces of powers of complex matrices of size $N$, in term of a determinant; this determinant is…
Various lattice path models are reviewed. The enumeration is done using generating functions. A few bijective considerations are woven in as well. The kernel method is often used. Computer algebra was an essential tool. Some results are…
A vector composition of a vector $\mathbf{\ell}$ is a matrix $\mathbf{A}$ whose rows sum to $\mathbf{\ell}$. We define a weighted vector composition as a vector composition in which the column values of $\mathbf{A}$ may appear in different…
We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves…
Let $A$ and $B$ be complex numbers, and let $(w_n)_{n\ge0}$ be a sequence of complex numbers with $w_{n+1}=Aw_n-Bw_{n-1}$ for all $n=1,2,3,\ldots$. When $w_0=0$ and $w_1=1$, the sequence $(w_n)_{n\ge0}$ is just the Lucas sequence…
In this article algebraic constructions are introduced in order to study the variety defined by a radical parametrization (a tuple of functions involving complex numbers, $n$ variables, the four field operations and radical extractions). We…
Motzkin paths of order-$\ell$ are a generalization of Motzkin paths that use steps $U=(1,1)$, $L=(1,0)$, and $D_i=(1,-i)$ for every positive integer $i \leq \ell$. We further generalize order-$\ell$ Motzkin paths by allowing for various…
In this paper, firstly, by a determinant of deformed Pascal's triangle, namely the normalized Hessenberg matrix determinant, to count Dyck paths, we give another combinatorial proof of the theorems which are of Catalan numbers determinant…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
We determine a family of functors from a poset to abelian groups such that the higher direct limits vanish on them. This is done by first characterizing the projective functors. Then a spectral sequence arising from the grading of the poset…
We present determinant formulae for the number of tilings of various domains in relation with Alternating Sign Matrix and Fully Packed Loop enumeration.
Combinatorial enumeration leads to counting generating functions presenting a wide variety of analytic types. Properties of generating functions at singularities encode valuable information regarding asymptotic counting and limit…
We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees).…
In this paper, we introduce two differential equations arising from the generating function of the Catalan numbers which are `inverses' to each other in some sense. From these differential equations, we obtain some new and explicit…
Meanders form a set of combinatorial problems concerned with the enumeration of self-avoiding loops crossing a line through a given number of points, $n$. Meanders are considered distinct up to any smooth deformation leaving the line fixed.…
Regular tree grammars and regular path expressions constitute core constructs widely used in programming languages and type systems. Nevertheless, there has been little research so far on reasoning frameworks for path expressions where node…
We define new families of noncommutative symmetric functions and quasi-symmetric functions depending on two matrices of parameters, and more generally on parameters associated with paths in a binary tree. Appropriate specializations of both…
In this expository paper we compute Hankel determinants of some sequences whose generating functions are given by C-fractions and derive orthogonality properties for associated polynomials.