Polyominoes, Permutominoes and Permutations Enrica Duchi

UniversitéParis Diderot { Ecole´ doctorale de sciences mathématiquesde Paris centre Mémoired'habilitation àdiriger des recherches SpécialitéInformatique Polyominoes, permutominoes and permutations Enrica Duchi Rapporteuses: Marilena Barnabei, Professeure, Universitàdi Bologna FrédériqueBassino, Professeure, UniversitéParis Nord ValérieBerthé, Directrice de recherche au CNRS Soutenue le 28 novembre 2018 àParis devant le jury composéde: FrédériqueBassino, Professeure, UniversitéParis Nord Fran¸coisBergeron, Professeur, Universitédu Québec àMontréal Jean-Marc Fédou, Professeur, Universitéde Nice Sophia-Antipolis Vlady Ravelomanana, Professeur, UniversitéParis Diderot Bruno Salvy, Directeur de recherche àl'INRIA MichèleSoria, Professeure, UniversitéPierre-et-Marie-Curie 2 Contents 1 Introduction and brief curriculum 5 I Convex polyominoes, convex permutominoes and square permutations: results and methods 15 2 Polyominoes, permutations and permutominoes 17 2.1 Convex polyominoes . 17 2.1.1 Polyominoes, polyominoes without holes, perimeter . 17 2.1.2 Convex polyominoes and their number, directed convex and parallel polyominoes . 18 2.1.3 Bibliographical notes on convex polyomino enumeration . 19 2.2 Square permutations . 20 2.2.1 Square permutations and their number . 20 2.2.2 Triangular and parallel permutations . 21 2.2.3 Square permutations vs convex polyominoes . 22 2.2.4 Bibliographical notes on square permutation enumeration . 22 2.3 Convex permutominoes . 22 2.3.1 Permutation diagrams and permutominoes . 22 2.3.2 Convex permutominoes and their number . 27 2.3.3 Convex permutominoes vs square permutation . 28 2.3.4 Bibliographical notes on convex permutomino enumeration . 29 2.4 Geometrical interpretations and asymptotical relations . 29 2.5 A summary and some questions . 31 3 Methods 34 3.1 The linear recursive approach . 34 3.1.1 Generating trees and succession rules . 34 3.1.2 A generating tree for parallelogram polyominoes . 35 3.1.3 A generating tree for convex permutominoes . 38 3.1.4 Two succession rules for convex polyominoes and square permutations . 41 3.1.5 Other linear recursive constructions for convex polyominoes and square permutations . 42 3.1.6 Some remarks about linear equation with one catalytic variable . 42 3.2 N-Algebraic decompositions . 43 3.2.1 Introduction to object grammars . 44 3.2.2 Grammars for Catalan objects . 45 3.2.3 Grammars for central binomial objects . 49 3.2.4 Some remarks about object grammars . 55 3.3 Direct bijections . 55 3.3.1 Bijections and the Catalan garden . 56 3.3.2 Bijections for central binomial structures . 60 3 3.3.3 Bijections for half central binomial coefficients . 66 3.3.4 Remarks about the bijective approach . 68 3.4 Enumeration by difference . 69 3.4.1 The enumeration of convex polyominoes by difference . 69 3.4.2 Enumeration of convex permutominoes by difference . 70 3.4.3 Enumeration of square permutations and convex permutominoes by difference 74 3.4.4 Some remarks about differences . 79 II Three examples of more advanced combinatorial constructions 80 4 Permutations with few internal points 82 4.1 The standard generating tree for permutations . 82 4.2 A new generating tree for permutations . 83 4.2.1 The generating tree for permutations without internal points . 83 4.2.2 Producing internal points . 86 4.3 The shape of the generating tree . 86 4.3.1 A classification of permutations and the related parameters . 87 4.3.2 One of several cases: the action of # on Class Dd;c .............. 88 4.4 Taking advantage of regularities . 90 4.4.1 Functional equations . 90 4.4.2 Enumerative results . 91 5 A Generating Tree for Permutations Avoiding the Pattern 122+3 93 5.1 Pattern avoidance . 93 5.2 Getting started . 94 5.3 Generation of AV (122+3)................................ 96 5.4 Generation of Dyck paths with marked valleys . 99 5.5 A bijection between AV (122+3) and Dyck paths with marked valleys . 101 6 Fighting fish 103 6.1 Introduction to fighting fish . 103 6.2 Enumeration of fighting fish according to their size and area . 105 6.2.1 A recursive definition . 106 6.2.2 Generating functions . 107 6.3 Refined generating function . 113 6.3.1 A wasp-waist decomposition . 113 6.3.2 The algebraic solution of the functional equation . 115 6.4 A refined conjecture . 116 4 Chapter 1 Introduction and brief curriculum Enumerative and bijective combinatorics Combinatorics is nowadays a fundamental research field representing one of the main bridges between computer science and mathematics. It deals with the mathematical properties of discrete structures, as opposed to continuous ones, and structures of this kind play an important role in theoretical computer science. Within combinatorics, enumerative combinatorics is more specifically dealing with the fundamental problem of counting structures in combinatorial classes with respect to a size, in an exact or in an approximate way. Apart the natural question to know how many objects there are, enumerative problems arises in many fields, ranging from the analysis of algorithms to statistical physics or bioinformatics for instance. Sometimes it is easy to count, like in the case of permutations of 1; : : : ; n , counted by n!, but in general this is not the case. Only in rare cases the answer willf be a completelyg explicit closed formula, involving well known functions, and free from summation symbols. Whenever we do not have a simple formula for counting numbers we can hope to get one for the generating function of the combinatorial class, and this is the reason why a lot of energy has been dedicated to improve the toolbox of enumerative combinatorics: clever decomposition technics leading to functional equations for generating functions, or sophisticated methods to solve these classes of functional equations, or at least to obtain asymptotic results. Still, sometimes these sophisticated approaches lead to surprisingly nice closed formulas or to unexpected coincidences when apparently unrelated objects reveal themselves equinumerous, then we ask for a direct combinatorial explanation of these facts: bijective combinatorics comes in this context. Let us take an example: the number of spanning trees of the complete graph with nodes in 1; : : : ; n , called Cayley trees, is nn−2 and it is also the number of different ways to write the cyclicf permutationg (1; 2; : : : ; n) into a product of n 1 transpositions. While the first one is a very classical result with a lot of different proofs and− generalizations, the second one is less known outside the combinatorics community but it attracts the attention because of the appearance of the same numbers in a priori different context. Such coincidences are at the basis of bijective combinatorics. Dénes,in the 50's, looked for an explanation for the coincidence of the nn−2 numbers, and obtained a first bijection between doubly labelled Cayley trees and factorizations of arbitrary big cycles in transpositions, then a second bijection directly between Cayley trees and factorizations of a fixed big cycle was given by Mozkovski in the 80's. On the one side Dénes bijection makes explicit the arborescent structures of the junction operations between cycles in a product of transpositions, on the other side Mozkovski's bijections has allowed to realize that Cayley trees can be seen as planar increasing trees, that is a relevant ingredient in a lot of later works, including some of mines. In general the fact of explaining a formula through a bijection lead us to obtain a better understanding of the properties of the underlying objects. 5 We have already pointed out that it is not easy to find closed formulas for combinatorial classes, so that when we have a method that works, either a decomposition or some manipulations on a system of functional equations, it is natural to wonder how far this method can be generalized. In particular if we have a non standard decomposition for a combinatorial class one can wonder if it is a decomposition ad hoc for that class or if it can apply to other combinatorial classes. This can be done by looking for general theorems formalizing the approach, but also, and maybe more importantly in combinatorics, by looking for alternative interesting examples: while general theorems are attractive, they are only useful with good applications, which are not so easy to find... Most of my research is set in this field of bijective and enumerative combinatorics, and is concerned with the search of good examples! A quick overview of my research In order to present in more details my field of research I have decided to concentrate on a coincidence of the type discussed above: the stricking similarity between classical Delest-Viennot formula for convex polyominoes and the much more recent formulas for square permutations and convex permutominoes. The later combinatorial classes and their counting formulas are much less known than polyominoes but deserve in my opinion some advertising! In a first part I will thus in- troduce these objects and show how some standard enumerative methods converge on their study. Then in a second part I will discuss some more advanced examples, dealing with generalizations of directed convex polyominoes and square permutations, and with a class of pattern avoiding permutations. In the rest of this preliminary chapter I will now review some topics I have been interested in over the recent years in approximate chronological order, just to show that polyominoes, permutominoes and permutations, although they play an important role in my work, are not the only combinatorial classes I have been considering... Along the lines, I will point out those works that will appear, briefly or in details, later in the manuscript. This classification will be followed by a short curriculum vitae.

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