Renzo Sprugnoli

Riordan arrays and combinatorial sums

Abstract The concept of a Riordan array is used in a constructive way to find the generating function of many combinatorial sums. The generating function can then help us to obtain the closed form of the sum or its asymptotic value. Some examples of sums involving binomial coefficients and Stirling numbers are examined, together with an application of Riordan arrays to some walk problems.

Sequence characterization of Riordan arrays

In the realm of the Riordan group, we consider the characterization of Riordan arrays by means of the A- and Z-sequences. It corresponds to a horizontal construction of a Riordan array, whereas the traditional approach is through column generating functions. We show how the A- and Z-sequences of the product of two Riordan arrays are derived from those of the two factors; similar results are obtained for the inverse. We also show how the sequence characterization is applied to construct easily a Riordan array. Finally, we give the characterizations relative to some subgroups of the Riordan group, in particular, of the hitting-time subgroup.

Riordan arrays and the Abel-Gould identity

Abstract We generalize the well-known identities of Abel and Gould in the context of Riordan arrays. This allows us to prove analogous formulas for Stirling numbers of both kinds and also for other quantities.

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.

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 random generation of directed animals

Abstract In this paper, we propose an algorithm to randomly generate a directed animal. Directed animals are well-known combinatorial objects and have been widely used for modelling the physical phenomenon of percolation. The algorithm consists of three steps, and we prove that each of them is performed in linear time. Finally, we report the results of our experiments made by means of appropriate computer programs in order to give empirical evidence that our algorithm really works.

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.