M. Cecilia Verri

The Cauchy numbers

We study many properties of Cauchy numbers in terms of generating functions and Riordan arrays and find several new identities relating these numbers with Stirling, Bernoulli and harmonic numbers. We also reconsider the Laplace summation formula showing some applications involving the Cauchy numbers.

Generating trees and proper Riordan arrays

Abstract We use an algebraic approach to study the connection between generating trees and proper Riordan Arrays deriving a theorem that, under suitable conditions, associates a Riordan Array to a generating tree and vice versa. Thus, we can use results from the theory of Riordan Arrays to study properties of generating trees. In particular, we can find, in a general and easy way, the generating functions counting the distribution of trees’ nodes at each level. The connection between Riordan Arrays and transfer matrices is also shown.

Some statistics on Dyck paths

We study some statistics related to Dyck paths, whose explicit formulas are obtained by means of the Lagrange Inversion Theorem. There are five such statistics and one of them is well-known and owed to Narayana. The most interesting of the other four statistics is related to Eulers trinomial coefficients and to Motzkin numbers: we perform a study of that statistic proving a number of its properties.

Underdiagonal lattice paths with unrestricted steps

Abstract We use some combinatorial methods to study underdiagonal paths (on the Z2 lattice) made up of unrestricted steps, i.e., ordered pairs of non-negative integers. We introduce an algorithm which automatically produces some counting generating functions for a large class of these paths. We also give an example of how we use these functions to obtain some specific information on the number dn, k of paths from the origin to a generic point (n, n − k).

The Area Determined by Underdiagonal Lattice Paths

We use the “first passage decomposition” methodology to study the area between various kinds of underdiagonal lattice paths and the main diagonal. This area is important because it is connected to the number of inversions in permutations and to the internal path length in various types of trees. We obtain the generating functions for the total area of all the lattice paths from the origin to the point (n, n). Since this method also determines the number of these paths, we are able to obtain exact results for the average area.

Some more properties of Catalan numbers

We want to illustrate some correspondences between Catalan numbers and combinatoric objects, such as plane walks, binary trees and some particular words. By means of under-diagonal walks, we give a combinatorial interpretation of the formula Cn = 1n+12nn defining Catalan numbers. These numbers also enumerate both words in a particular language defined on a four character alphabet and the corresponding walks made up of four different types of steps. We illustrate a bijection between n-long words in this language and binary trees having n + 1 nodes, after which we give a simple proof of Touchards formula.

Algebraic and Combinatorial Properties of Simple, Coloured Walks

We investigate the algebraic rules for functionally inverting a Riordan array given by means of two analytic functions. In this way, we find an extension of the Lagrange Inversion Formula and we apply it to some combinatorial problems on simple coloured walks. For some of these problems we give both an algebraic and a combinatorial proof.

Strip tiling and regular grammars

We study the problem of tiling a rectangular p×n-strip ( fixed, ) with pieces, i.e., sets of simply connected cells. Some well-known examples are strip tilings with dimers (dominoes) and/or monomers. We prove, in a constructive way, that every tiling problem is equivalent to a regular grammar, that is, the set of possible tilings constitutes a regular language. We propose a straightforward algorithm to transform the tiling problem into its corresponding grammar. By means of some standard methods, we are then able to obtain some counting generating functions that are rational. We go on to give some examples of our method and indicate some of its applications to a number of problems treated in current literature.

A uniform model for the storage utilization of B-tree-like structures

By using the method of Riordan arrays we find the asymptotic value of the expected storage utilization in a model of uniform B-tree-like structures proposed by Gupta and Srinivasan, thus solving the occupancy problem for that model.

An Universal Termination Condition for Solving Goals in Equational Languages

A decomposition procedure, called DP, operating on a ”sorted” set of equations is here used as the operational semantics of CTRS. Then, a class of CTRS called conic-flat, is defined for which DP is shown to be universally terminating when solving the equation t1=Rt2, with t1 or t2 ground.