We show that the generating function of the symmetric group with respect to five particular statistics gives rise to an exponential Riordan array, whose inverse is the coefficient array of the associated orthogonal polynomials. This also provides us with an LDU factorization of the Hankel matrix of the associated moments.

The Chebyshev-Boubaker polynomials are the orthogonal polynomials whose coefficient arrays are defined by ordinary Riordan arrays. Examples include the Chebyshev polynomials of the second kind and the Boubaker polynomials. We study the connection coefficients of this class of orthogonal polynomials, indicating how Riordan array techniques can lead to closed-form expressions for these connection...

Using the theory of exponential Riordan arrays and orthogonal polynomials, we demonstrate that the Eulerian polynomials and the shifted Eulerian polynomials are moment sequences for a simple family of orthogonal polynomials. The coefficient arrays of these families of orthogonal polynomials are shown to be exponential Riordan arrays. Using the theory of orthogonal polynomials we are then able t...

Bollobás and Riordan introduced a three-variable polynomial extending the Tutte polynomial to oriented ribbon graphs, which are multi-graphs embedded in oriented surfaces, such that complementary regions (faces) are discs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. By generalizing Tutte’s concept of activity to quasi-tree...

We give several new characterizations of Riordan Arrays, the most important of which is: if fd n;k g n;k2N is a lower triangular array whose generic element d n;k linearly depends on the elements in a well-deened though large area of the array, then fd n;k g n;k2N is Riordan. We also provide some applications of these characterizations to the lattice path theory.

One of the most fundamental results concerning Riordan arrays is that they have a sequence characterization [13, 18]. This normally involves two sequences, called the A-sequence and the Z-sequence. For exponential Riordan arrays [9] (see Appendix), this characterization is equivalent to the fact that the production matrix [11] of an exponential array [g, f ], with A-sequence A(t) and Z-sequence...

Ribbon graphs are surfaces with boundary together with a decomposition into a union of closed topological discs of two types, edges and vertices. These sets are subject to some natural axioms recalled in section 2.1. For such a generalisation of the usual graphs, B. Bollobás and O. Riordan found a topological version of the Tutte polynomial [3, 4]. In the following, we will refer to this genera...