Search

We consider sorting procedures for permutations making use of pop stacks with a bypass operation, and explore the combinatorial properties of the associated algorithms., Comment: In Proceedings GASCom 2024, arXiv:2406.14588

The conference GASCom brings together researchers in combinatorics, algorithms, probabilities, and more generally mathematical computer science, around the theme of random and exhaustive generation of combinatorial structures, mostly considered from a theoretical point of view. In connection with this main theme, the conference is also interested in contributions in enumerative or analytic combinatorics, and interactions with other areas of mathematics, computer science, physics or biology. The conference is both interested in methods for random or exhaustive generation and in original results on combinatorial or algorithmic questions, whose solution has been made possible by an approach involving random or exhaustive generation. The present edition of the conference includes a specific bunch of talks dedicated to polyominoes and tilings. A (not exhaustive) list of topics of the conference is: random and exhaustive generation of combinatorial objects; tilings and polyominoes; bijective, enumerative, algebraic and analytic combinatorics; algorithmic aspects: analysis of algorithms, probabilistic algorithms; interactions: bio-informatics, combinatorics on words, number theory.

We define a partial order $\mathcal{P}_n$ on permutations of any given size $n$, which is the image of a natural partial order on inversion sequences. We call this the ``middle order''. We demonstrate that the poset $\mathcal{P}_n$ refines the weak order on permutations and admits the Bruhat order as a refinement, justifying the terminology. These middle orders are distributive lattices and we establish some of their combinatorial properties, including characterization and enumeration of intervals and boolean intervals (in general, or of any given rank), and a combinatorial interpretation of their Euler characteristic. We further study the (not so well-behaved) restriction of this poset to involutions, obtaining a simple formula for the M\"obius function of principal order ideals there. Finally, we offer further directions of research, initiating the study of the canonical Heyting algebra associated with $\mathcal{P}_n$, and defining a parking function analogue of $\mathcal{P}_n$., Comment: 21 pages, 3 figures

Naples parking functions were introduced as a generalization of classical parking functions, in which cars are allowed to park backwards, by checking up to a fixed number of previous spots, before proceeding forward as usual. In this work we introduce the notion of a complete parking preference, through which we are able to give some information on the combinatorics of Naples parking functions. Roughly speaking, a complete parking preference is a parking preference such that, for any index $j$, there are more cars with preference at least $j$ than spots available from $j$ onward. We provide a characterization of Naples parking functions in terms of certain complete subsequences of them. As a consequence of this result we derive a characterization of permutation-invariant Naples parking functions which turns out to be equivalent to the one given by (Carvalho et al., 2021), but using a totally different approach (and language)., Comment: 12 pages

We study preimages of permutations under the bubblesort operator $\mathbf{B}$. We achieve a description of these preimages much more complete than what is known for the more complicated sorting operators $\mathbf{S}$ (stacksort) and $\mathbf{Q}$ (queuesort). We describe explicitly the set of preimages under $\mathbf{B}$ of any permutation $\pi$ from the left-to-right maxima of $\pi$, showing that there are $2^{k-1}$ such preimages if $k$ is the number of these left-to-right maxima. We further consider, for each $n$, the tree $T_n$ recording all permutations of size $n$ in its nodes, in which an edge from child to parent corresponds to an application of $\mathbf{B}$ (the root being the identity permutation), and we present several properties of these trees. In particular, for each permutation $\pi$, we show how the subtree of $T_n$ rooted at $\pi$ is determined by the number of left-to-right maxima of $\pi$ and the length of the longest suffix of left-to-right maxima of $\pi$. Building on this result, we determine the number of nodes and leaves at every height in such trees, and we recover (resp. obtain) the average height of nodes (resp. leaves) in $T_n$.

Using a recursive approach, we show that the generating function for sets of Motzkin paths avoiding a single (not necessarily consecutive) pattern is rational over $x$ and the Catalan generating function $C(x) = \frac{1-\sqrt{1-4x^2}}{2x^2}$, where $x$ keeps track of the length of the path. Moreover, an algorithm is provided for finding the generating function in the more general case of an arbitrary set of patterns. In addition, this algorithm allows us to find a combinatorial specification for pattern-avoiding Motzkin paths, which can be used not only for enumeration, but also for exhaustive and random generation.

Following the footprints of what have been done with the algorithm Stacksort, we investigate the preimages of the map associated with a slightly less well known algorithm, called Queuesort. After having described an equivalent version of Queuesort, we provide a recursive description of the set of all preimages of a given permutation, which can be also translated into a recursive procedure to effectively find such preimages. We then deal with some enumerative issues. More specifically, we investigate the cardinality of the set of preimages of a given permutation, showing that all cardinalities are possible, except for 3. We also give exact enumeration results for the number of permutations having 0,1 and 2 preimages. Finally, we consider the special case of those permutations $\pi$ whose set of left-to-right maxima is the disjoint union of a prefix and a suffix of $\pi$: we determine a closed formula for the number of preimages of such permutations, which involves two different incarnations of ballot numbers, and we show that our formula can be expressed as a linear combination of Catalan numbers., Comment: 18 pages

A matching of the set $[2n]=\{ 1,2,\ldots ,2n\}$ is a partition of $[2n]$ into blocks with two elements, i.e. a graph on $[2n]$ such that every vertex has degree one. Given two matchings $\sigma$ and $\tau$ , we say that $\sigma$ is a pattern of $\tau$ when $\sigma$ can be obtained from $\tau$ by deleting some of its edges and consistently relabelling the remaining vertices. This is a partial order relation turning the set of all matchings into a poset, which will be called the matching pattern poset. In this paper, we continue the study of classes of pattern avoiding matchings, initiated by Chen, Deng, Du, Stanley and Yan (2007), Jelinek and Mansour (2010), Bloom and Elizalde (2012). In particular, we work out explicit formulas to enumerate the class of matchings avoiding two new patterns, obtained by juxtaposition of smaller patterns, and we describe a recursive formula for the generating function of the class of matchings avoiding the lifting of a pattern and two additional patterns. Finally, we introduce the notion of unlabeled pattern, as a combinatorial way to collect patterns, and we provide enumerative formulas for two classes of matchings avoiding an unlabeled pattern of order three. In one case, the enumeration follows from an interesting bijection between the matchings of the class and ternary trees., Comment: 18 pages, presented at the 2020 Virtual Workshop on Permutation Patterns

This paper continues the analysis of the pattern-avoiding sorting machines recently introduced by Cerbai, Claesson and Ferrari [CCF]. These devices consist of two stacks, through which a permutation is passed in order to sort it, where the content of each stack must at all times avoid a certain pattern. Here we characterize and enumerate the set of permutations that can be sorted when the first stack is $132$-avoiding, solving one of the open problems proposed in [CCF]. To that end we present several connections with other well known combinatorial objects, such as lattice paths and restricted growth functions (which encode set partitions). We also provide new proofs for the enumeration of some sets of pattern-avoiding restricted growth functions and we expect that the tools introduced can be fruitfully employed to get further similar results., Comment: 25 pages, 7 figures

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., Comment: 16 pages, 4 figures

We initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the M\"obius function of the Dyck pattern poset, giving for instance a closed expression for the M\"obius function of initial intervals whose maximum is a Dyck path having exactly two peaks., Comment: 14 pages, 3 figures

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 $\sigma$-avoiding, for some permutation $\sigma$, meaning that, at each step, the elements maintained in the stack avoid the pattern $\sigma$ when read from top to bottom. Since the set of permutations which can be sorted by such a device (which we call $\sigma$-machine) is not always a class, it would be interesting to understand when it happens. We will prove that the set of $\sigma$-machines whose associated sortable permutations are not a class is counted by Catalan numbers. Moreover, we will analyze two specific $\sigma$-machines in full details (namely when $\sigma=321$ and $\sigma=123$), providing for each of them a complete characterization and enumeration of sortable permutations.

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., Comment: 12 pages

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 $k$., Comment: 20 pages, 1 figure

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 $k$., Comment: 8 pages. In: L. Ferrari, M. Vamvakari (eds.): Proceedings of the GASCom 2018 Workshop, Athens, Greece, 18--20 June 2018, published at http://ceur-ws.org

The set of Schr\"oder words (Schr\"oder language) is endowed with a natural partial order, which can be conveniently described by interpreting Schr\"oder words as lattice paths. The resulting poset is called the Schr\"oder pattern poset. We find closed formulas for the number of Schr\"oder words covering/covered by a given Schr\"oder word in terms of classical parameters of the associated Schr\"oder path. We also enumerate several classes of Schr\"oder avoiding words (with respect to the length), i.e. sets of Schr\"oder words which do not contain a given Schr\"oder word., Comment: 12 pages, 1 figure

Let h : R $\rightarrow$ R+ be a lower semi-continuous subbadditive and even function such that h(0) = 0 and h($\theta$) $\ge$ $\alpha$|$\theta$| for some $\alpha$ > 0. The h-mass of a k-polyhedral chain P =$\sum$j $\theta$j$\sigma$j in R n (0 $\le$ k $\le$ n) is defined as M h (P) := j h($\theta$j) H k ($\sigma$j). If T = $\tau$ (M, $\theta$, $\xi$) is a k-rectifiable chain, the definition extends to M h (T) := M h($\theta$) dH k. Given such a rectifiable flat chain T with M h (T) < $\infty$ and $\partial$T polyhedral, we prove that for every $\eta$ > 0, it decomposes as T = P + $\partial$V with P polyhedral, V rectifiable, M h (V) < $\eta$ and M h (P) < M h (T) + $\eta$. In short, we have a polyhedral chain P which strongly approximates T in h-mass and preserves the homological constraint $\partial$P = $\partial$T. These results are motivated by the study of approximations of M h by smoother functionals but they also provide explicit formulas for the lower semicontinuous envelope of T $\rightarrow$ M h (T) + I $\partial$S ($\partial$T) with respect to the topology of the flat norm.

In branched transportation problems mass has to be transported from a given initial distribution to a given final distribution, where the cost of the transport is proportional to the transport distance, but subadditive in the transported mass. As a consequence, mass transport is cheaper the more mass is transported together, which leads to the emergence of hierarchically branching transport networks. We here consider transport costs that are piecewise affine in the transported mass with N affine segments, in which case the resulting network can be interpreted as a street network composed of N different types of streets. In two spatial dimensions we propose a phase field approximation of this street network using N phase fields and a function approximating the mass flux through the network. We prove the corresponding $\Gamma$-convergence and show some numerical simulation results.

In this paper we produce a $$\Gamma$$-convergence result for a class of energies $F k $\epsilon$,a$ modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that $F 1 $\epsilon$,a $\Gamma$$-converges to a branched transportation energy whose cost per unit length is a function $f n--1 a$ depending on a parameter $a > 0$ and on the codimension n -- 1. The limit cost f a (m) is bounded from below by 1 + m so that the limit functional controls the mass and the length of the limit object. In the limit a $\downarrow$ 0 we recover the Steiner energy. We then generalize the approach to any dimension and codimension. The limit objects are now k-currents with prescribed boundary, the limit functional controls both their masses and sizes. In the limit $a $\downarrow$ 0$, we recover the Plateau energy defined on k-currents, $k < n$. The energies $F k $\epsilon$,a$ then can be used for the numerical treatment of the k-Plateau problem., Comment: ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, In press