Enrica Duchi

On the equivalence problem for succession rules

The notion of succession rule (system for short) provides a powerful tool for the enumeration of many classes of combinatorial objects. Often, different systems exist for a given class of combinatorial objects, and a number of problems arise naturally. An important one is the equivalence problem between two different systems. In this paper, we show how to solve this problem in the case of systems having a particular form. More precisely, using a bijective proof, we show that the classical system defining the sequence of Catalan numbers is equivalent to a system obtained by linear combinations of labels of the first one.

Bijective Construction of Equivalent Eco-systems

First we explicit an infinite family of equivalent succession rules parametrized by a positive integer α, for which two specializations lead to the equivalence of rules defining the Catalan and Schroder numbers. Then, from an ECO-system for Dyck paths easily derive an ECO-system for complete binary trees y using a widely known bijection between these objects. We also give a similar construction in the less easy case of Schroder paths and Schroder trees which generalizes the previous one.

Permutations with few internal points

Abstract Let the records of a permutation σ be its left-right minima, right-left minima, left-right maxima and right-left maxima. Conversely let a point ( i , j ) with j = σ ( i ) be an internal point of σ if it is not a record. Permutations without internal points have recently attracted attention under the name square permutations. We consider here the enumeration of permutations with a fixed number of internal points. We show that for each fixed i the generating function of permutations with i internal points with respect to the size is algebraic of degree 2. More precisely it is a rational function in the Catalan generating function. Our approach is constructive, yielding a polynomial uniform random sampling algorithm, and it can be refined to enumerate permutations with respect to each of the four types of records.