Donatella Merlini

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.

Left-inversion of combinatorial sums

Abstract The inversion of combinatorial sums is a fundamental problem in algebraic combinatorics. Some combinatorial sums, such as a n = Σ k d n,k b k , cannot be inverted in terms of the orthogonality relation because the infinite, lower triangular array P = { d n,k }s diagonal elements are equal to zero (except d 0,0 ). Despite this, we can find a left-inverse P such that PP = I and therefore are able to left-invert the original combinatorial sum, and thus obtain b n = Σ k d n,k a k .

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.

The Tennis Ball Problem

Abstract Mallows and Shapiro, ( J. Integer Sequences 2 (1999)) have recently considered what they dubbed the problem of balls on the lawn . Our object is to explore a natural generalization, the s-tennis ball problem , which reduces to that considered by Mallows and Shapiro in the case s =2. We show how this generalization is connected with s -ary trees, and employ the notion of generating trees to obtain a solution expressed in terms of generating functions.

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).

Data mining models for student careers

We presents a data mining methodology to analyze the careers of University graduated students.We present different approaches based on clustering and sequential patterns techniques.We introduce the concept of ideal career.We compare the career of a generic student with the ideal one.We apply the methodology to a real case study and interpret the results. This paper presents a data mining methodology to analyze the careers of University graduated students. We present different approaches based on clustering and sequential patterns techniques in order to identify strategies for improving the performance of students and the scheduling of exams. We introduce an ideal career as the career of an ideal student which has taken each examination just after the end of the corresponding course, without delays. We then compare the career of a generic student with the ideal one by using the different techniques just introduced. Finally, we apply the methodology to a real case study and interpret the results which underline that the more students follow the order given by the ideal career the more they get good performance in terms of graduation time and final grade.

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.

Waiting patterns for a printer

We introduce a model based on some combinatorial objects, which we call l-histograms, to study the behaviour of devices like printers and use the combinatorial properties of these objects to study some important distributions such as the waiting time for a job and the length of the device queue. This study is based on an important relation between 1-histograms, generating trees and binary trees.

Algebraic aspects of some Riordan arrays related to binary words avoiding a pattern

We consider some Riordan arrays related to binary words avoiding a pattern p, which can be easily studied by means of an A-matrix rather than their A-sequence. Both concepts allow us to define every element as a linear combination of other elements in the array; the A-sequence is unique and corresponds to a linear dependence from the previous row. The A-matrix is not unique and corresponds to a linear dependence from several previous rows. However, for the problems considered in the present paper, we show that the A-matrix approach is more convenient. We provide explicit algebraic generating functions for these Riordan arrays and obtain many statistics on the corresponding languages. We thus obtain a deeper insight of the languages L^[^p^] of binary words avoiding p having a number of 0s less or equal to the number of 1s.