96 research outputs found
Simple proofs of open problems about the structure of involutions in the Riordan group
AbstractWe prove that if D=(g(x),f(x)) is an element of order 2 in the Riordan group then g(x)=±exp[Φ(x,xf(x)] for some antisymmetric function Φ(x,z). Also we prove that every element of order 2 in the Riordan group can be written as BMB-1 for some element B and M=(1,-1) in the Riordan group. These proofs provide solutions to two open problems presented by L. Shapiro [L.W. Shapiro, Some open questions about random walks, involutions, limiting distributions and generating functions, Adv. Appl. Math. 27 (2001) 585–596]
The uplift principle for ordered trees
AbstractIn this paper, we describe the uplift principle for ordered trees which lets us solve a variety of combinatorial problems in two simple steps. The first step is to find the appropriate generating function at the root of the tree, the second is to lift the result to an arbitrary vertex by multiplying by the leaf generating function. This paper, though self contained, is a companion piece to Cheon and Shapiro (2008) [2] though with many more possible applications. It also may be viewed as an invitation, via the symbolic method, to the authoritative 800 page book of Flajolet and Sedgewick (2009) [8]. Our examples, with one exception, are different from those in this excellent reference
Protected points in ordered trees
AbstractIn this note we start by computing the average number of protected points in all ordered trees with n edges. This can serve as a guide in various organizational schemes where it may be desirable to have a large or small number of protected points. We will also look a few subclasses with a view to increasing or decreasing the proportion of protected points
On naturally labelled posets and permutations avoiding 12-34
A partial order on is naturally labelled (NL) if
implies . We establish a bijection between {3, 2+2}-free NL posets and
12-34-avoiding permutations, determine functional equations satisfied by their
generating function, and use series analysis to investigate their asymptotic
growth. We also exhibit bijections between 3-free NL posets and various other
objects, and determine their generating function. The connection between our
results and a hierarchy of combinatorial objects related to interval orders is
described.Comment: 19 page
Rook polynomials to and from permanents
AbstractIn this paper, we find an expression of the rook vector of a matrix A (not necessarily square) in terms of permanents of some matrices associated with A, and obtain some simple exact formulas for the permanents of all n×n Toeplitz band matrices of zeros and ones whose bands are of width not less than n−1
Riordan graphs I : structural properties
In this paper, we use the theory of Riordan matrices to introduce the notion of a Riordan graph. The Riordan graphs are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other fami- lies of graphs. The Riordan graphs are proved to have a number of interesting (fractal) properties, which can be useful in creating computer networks with certain desirable features, or in obtaining useful information when designing algorithms to compute values of graph invariants. The main focus in this paper is the study of structural properties of families of Riordan graphs obtained from infinite Riordan graphs, which includes a fundamental decomposition theorem and certain conditions on Riordan graphs to have an Eulerian trail/cycle or a Hamiltonian cycle. We will study spectral properties of the Riordan graphs in a follow up paper
Riordan graphs II : spectral properties
The authors of this paper have used the theory of Riordan matrices to introduce the notion of a Riordan graph in [3]. Riordan graphs are proved to have a number of interesting (fractal) properties, and they are a far-reaching generalization of the well known and well studied Pascal graphs and Toeplitz graphs, and also some other families of graphs. The main focus in [3] is the study of structural properties of families of Riordan graphs obtained from certain infinite Riordan graphs. In this paper, we use a number of results in [3] to study spectral properties of Riordan graphs. Our studies include, but are not limited to the spectral graph invariants for Riordan graphs such as the adjacency eigenvalues, (signless) Laplacian eigenvalues, nullity, positive and negative inertia indices, and rank. We also study determinants of Riordan graphs, in particular, giving results about determinants of Catalan graphs
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