arXiv subjects

Giulio Cerbai

Publications and source records attributed to Giulio Cerbai.

22 records · Page 2Linked to original sources

arXiv2019

Stack Sorting with Increasing and Decreasing Stacks

We introduce a sorting machine consisting of $k+1$ stacks in series: the first $k$ stacks can only contain elements in decreasing order from top to bottom, while the last one has the opposite restriction. This device generalizes \cite{SM}, which studies the case $k=1$. Here we show that, for $k=2$, the set of sortable permutations is a class with infinite basis, by explicitly finding an antichain of minimal nonsortable permutations. This construction can easily be adapted to each $k \ge 3$. Next we describe an optimal sorting algorithm, again for the case $k=2$. We then analyze two types of left-greedy sorting procedures, obtaining complete results in one case and only some partial results in the other one. We close the paper by discussing a few open questions.

cs.DS↗

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↗