arXiv subjects

Luca Ferrari

Publications and source records attributed to Luca Ferrari.

At least 37 records · Page 2Linked to original sources

arXiv2019

Enhancing the connections between patterns in permutations and forbidden configurations in restricted elections

We investigate the connections between patterns in permutations and forbidden configurations in restricted elections, first discovered by Lackner and Lackner, in order to enhance the approach initiated by the two mentioned authors. More specifically, our achievements are essentially two. First, we define a new type of domain restriction, called enriched group-separable. Enriched group-separable elections are a subset of group-separable elections, which describe a special, still natural, situation that can arise in the context of group-separability. The exact enumeration of group-separable elections has been very recently determined by Karpov. Here we give a recursive characterization for enriched group-separable elections, from which we are able to find a recurrence relation and a closed formula expressing their number. Our second achievement is a generalization of a result of Lackner and Lackner, concerning the connection between permutation patterns and forbidden configurations with 3 voters. Our result relates forbidden configurations with the strong order on pairs of permutations, a notion which is still largely undeveloped, and suggests a potential approach for the determination of upper bounds for restricted elections whose forbidden configurations contains at least one configuration with 3 voters.

math.CO↗

arXiv2019

Stack sorting with restricted stacks

The (classical) problem of characterizing and enumerating permutations that can be sorted using two stacks connected in series is still largely open. In the present paper we address a related problem, in which we impose restrictions both on the procedure and on the stacks. More precisely, we consider a greedy algorithm where we perform the rightmost legal operation (here "rightmost" refers to the usual representation of stack sorting problems). Moreover, the first stack is required to be $σ$-avoiding, for some permutation $σ$, meaning that, at each step, the elements maintained in the stack avoid the pattern $σ$ when read from top to bottom. Since the set of permutations which can be sorted by such a device (which we call $σ$-machine) is not always a class, it would be interesting to understand when it happens. We will prove that the set of $σ$-machines whose associated sortable permutations are not a class is counted by Catalan numbers. Moreover, we will analyze two specific $σ$-machines in full details (namely when $σ=321$ and $σ=123$), providing for each of them a complete characterization and enumeration of sortable permutations.

cs.DS↗

arXiv2019

Permutation patterns in genome rearrangement problems: the reversal model

In the context of the genome rearrangement problem, we analyze two well known models, namely the reversal and the prefix reversal models, by exploiting the connection with the notion of permutation pattern. More specifically, for any $k$, we provide a characterization of the set of permutations having distance $\leq k$ from the identity (which is known to be a permutation class) in terms of what we call generating peg permutations and we describe some properties of its basis, which allow to compute such a basis for small values of $k$.

math.CO↗

arXiv2018

Permutation patterns in genome rearrangement problems

In the context of the genome rearrangement problem, we analyze two well known models, namely the block transposition and the prefix block transposition models, by exploiting the connection with the notion of permutation pattern. More specifically, for any $k$, we provide a characterization of the set of permutations having distance $\leq k$ from the identity (which is known to be a permutation class) in terms of what we call generating permutations and we describe some properties of its basis, which allow to compute such a basis for small values of $k$.

math.CO↗

arXiv2018

Enumerative Results on the Schröder Pattern Poset

The set of Schröder words (Schröder language) is endowed with a natural partial order, which can be conveniently described by interpreting Schröder words as lattice paths. The resulting poset is called the Schröder pattern poset. We find closed formulas for the number of Schröder words covering/covered by a given Schröder word in terms of classical parameters of the associated Schröder path. We also enumerate several classes of Schröder avoiding words (with respect to the length), i.e. sets of Schröder words which do not contain a given Schröder word.

math.CO↗

arXiv2018

Schröder partitions, Schröder tableaux and weak poset patterns

We introduce the notions of Schröder shape and of Schröder tableau, which provide some kind of analogs of the classical notions of Young shape and Young tableau. We investigate some properties of the partial order given by containment of Schröder shapes. Then we propose an algorithm which is the natural analog of the well known RS correspondence for Young tableaux, and we characterize those permutations whose insertion tableaux have some special shapes. The last part of the article relates the notion of Schröder tableau with those of interval order and of weak containment (and strong avoidance) of posets. We end our paper with several suggestions for possible further work.

math.CO↗

arXiv2015

Vincular pattern posets and the Möbius function of the quasi-consecutive pattern poset

We introduce vincular pattern posets, then we consider in particular the quasi-consecutive pattern poset, which is defined by declaring $σ\leq τ$ whenever the permutation $τ$ contains an occurrence of the permutation $σ$ in which all the entries are adjacent in $τ$ except at most the first and the second. We investigate the Möbius function of the quasi-consecutive pattern poset and we completely determine it for those intervals $[σ,τ]$ such that $σ$ occurs precisely once in $τ$.

math.CO↗

arXiv2015

Dyck algebras, interval temporal logic and posets of intervals

We investigate a natural Heyting algebra structure on the set of Dyck paths of the same length. We provide a geometrical description of the operations of pseudocomplement and relative pseudocomplement, as well as of regular elements. We also find a logic-theoretic interpretation of such Heyting algebras, which we call Dyck algebras, by showing that they are the algebraic counterpart of a certain fragment of a classical interval temporal logic (also known as Halpern-Shoham logic). Finally, we propose a generalization of our approach, suggesting a similar study of the Heyting algebra arising from the poset of intervals of a finite poset using Birkhöff duality. In order to illustrate this, we show how several combinatorial parameters of Dyck paths can be expressed in terms of the Heyting algebra structure of Dyck algebras together with a certain total order on the set of atoms of each Dyck algebra.

math.CO↗

arXiv2013

Greedy algorithms and poset matroids

We generalize the matroid-theoretic approach to greedy algorithms to the setting of poset matroids, in the sense of Barnabei, Nicoletti and Pezzoli (1998) [BNP]. We illustrate our result by providing a generalization of Kruskal algorithm (which finds a minimum spanning subtree of a weighted graph) to abstract simplicial complexes.

math.CO↗

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↗

arXiv2012

Unimodality and Dyck paths

We propose an original approach to the problem of rankunimodality for Dyck lattices. It is based on a well known recursive construction of Dyck paths originally developed in the context of the ECO methodology, which provides a partition of Dyck lattices into saturated chains. Even if we are not able to prove that Dyck lattices are rank-unimodal, we describe a family of polynomials (which constitutes a polynomial analog of ballot numbers) and a succession rule which appear to be useful in addressing such a problem. At the end of the paper, we also propose and begin a systematic investigation of the problem of unimodality of succession rules.

math.CO↗

arXiv2012

Enumeration of edges in some lattices of paths

We enumerate the edges in the Hasse diagram of several lattices arising in the combinatorial context of lattice paths. Specifically, we will consider the case of Dyck, Grand Dyck, Motzkin, Grand Motzkin, Schröder and Grand Schröder lattices. Finally, we give a general formula for the number of edges in an arbitrary Young lattice (which can be interpreted in a natural way as a lattice of paths).

math.CO↗

arXiv2012

Enumeration of saturated chains in Dyck lattices

We determine a general formula to compute the number of saturated chains in Dyck lattices, and we apply it to find the number of saturated chains of length 2 and 3. We also compute what we call the Hasse index (of order 2 and 3) of Dyck lattices, which is the ratio between the total number of saturated chains (of length 2 and 3) and the cardinality of the underlying poset.

math.CO↗

arXiv2012

A partial order structure on interval orders

We introduce a partial order structure on the set of interval orders of a given size, and prove that such a structure is in fact a lattice. We also provide a way to compute meet and join inside this lattice. Finally, we show that, if we restrict to series parallel interval order, what we obtain is the classical Tamari 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↗

arXiv2010

On the enumeration of d-minimal permutations

We suggest an approach for the enumeration of minimal permutations having d descents which uses skew Young tableaux. We succeed in finding a general expression for the number of such permutations in terms of (several) sums of determinants. We then generalize the class of skew Young tableaux under consideration; this allows in particular to discover some presumably new results concerning Eulerian numbers.

math.CO↗

arXiv2010

Catalan lattices on series parallel interval orders

Using the notion of series parallel interval order, we propose a unified setting to describe Dyck lattices and Tamari lattices (two well known lattice structures on Catalan objects) in terms of basic notions of the theory of posets. As a consequence of our approach, we find an extremely simple proof of the fact that the Dyck order is a refinement of the Tamari one. Moreover, we provide a description of both the weak and the strong Bruhat order on 312-avoiding permutations, by recovering the proof of the fact that they are isomorphic to the Tamari and the Dyck order, respectively; our proof, which simplifies the existing ones, relies on our results on series parallel interval orders.

math.CO↗

arXiv2009

Lattices of Paths: Representation Theory and Valuations

We study some distributive lattices arising in the combinatorics of lattice paths. In particular, for the Dyck, Motzkin and Schroder lattices we describe the spectrum and we determine explicitly the Euler characteristic in terms of natural parameters of lattice paths.

math.CO↗