arXiv subjects

Antonio Bernini

Publications and source records attributed to Antonio Bernini.

24 records · Page 2Linked to original sources

arXiv2013

The Dyck pattern poset

We introduce the notion of pattern in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the Dyck pattern poset. Given a Dyck path P, we determine a formula for the number of Dyck paths covered by P, as well as for the number of Dyck paths covering P. We then address some typical pattern-avoidance issues, enumerating some classes of pattern-avoiding Dyck paths. Finally, we offer a conjecture concerning the asymptotic behavior of the sequence counting Dyck paths avoiding a generic pattern and we pose a series of open problems regarding the structure of the Dyck pattern poset.

math.CO↗

arXiv2011

The Möbius function of the consecutive pattern poset

An occurrence of a consecutive permutation pattern $p$ in a permutation $π$ is a segment of consecutive letters of $π$ whose values appear in the same order of size as the letters in $p$. The set of all permutations forms a poset with respect to such pattern containment. We compute the Möbius function of intervals in this poset, providing what may be called a complete solution to the problem. For most intervals our results give an immediate answer to the question. In the remaining cases, we give a polynomial time algorithm to compute the Möbius function. In particular, we show that the Möbius function only takes the values -1, 0 and 1.

math.CO↗

arXiv2007

Enumeration of some classes of words avoiding two generalized patterns of length three

The method we have applied in "A. Bernini, L. Ferrari, R. Pinzani, Enumerating permutations avoiding three Babson-Steingrimsson patterns, Ann. Comb. 9 (2005), 137--162" to count pattern avoiding permutations is adapted to words. As an application, we enumerate several classes of words simultaneously avoiding two generalized patterns of length 3.

math.CO↗

arXiv2007

Enumerating permutations avoiding more than three Babson - Steingr\'ımsson patterns

Not long ago, Claesson and Mansour proposed some conjectures about the enumeration of the permutations avoiding more than three Babson - Steingr\'ımsson patterns (generalized patterns of type $(1,2)$ or $(2,1)$). The avoidance of one, two or three patterns has already been considered. Here, the cases of four and five forbidden patterns are solved and the exact enumeration of the permutations avoiding them is given, confirming the conjectures of Claesson and Mansour. The approach we use can be easily extended to the cases of more than five forbidden patterns.

math.CO↗

arXiv2007

A general exhaustive generation algorithm for Gray structures

Starting from a succession rule for Catalan numbers, we define a procedure encoding and listing the objects enumerated by these numbers such that two consecutive codes of the list differ only for one digit. Gray code we obtain can be generalized to all the succession rules with the stability property: each label $(k)$ has in its production two labels $c_1$ and $c_2$, always in the same position, regardless of $k$. Because of this link, we define Gray structures the sets of those combinatorial objects whose construction can be encoded by a succession rule with the stability property. This property is a characteristic that can be found among various succession rules, as the finite, factorial or transcendental ones. We also indicate an algorithm which is a very slight modification of the Walsh's one, working in a O(1) worst-case time per word for generating Gray codes.

math.CO↗

arXiv2007

An exhaustive generation algorithm for Catalan objects and others

In this paper we present a CAT generation algorithm for Dyck paths with a fixed length n. It is the formalization of a method for the exhaustive generation of this kind of paths which can be described by means of two equivalent strategies. The former is described by a rooted tree, the latter lists the paths by means of three operations which, as we are going to see, are equivalent to visit the tree. These constructions are strictly connected with ECO method and can be encoded by a rule, very similar to the succession rule in ECO, with a finite number of labels for each n. Moreover with a slight variation this method can be generalized to other combinatorial classes like Grand Dyck or Motzkin paths.

math.CO↗