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On Enumeration Sequences Generalising Catalan and Baxter Numbers department of information engineering and mathematics On enumeration sequences generalising Catalan and Baxter numbers Author: Veronica Guerrini Advisor: Prof. Simone Rinaldi, University of Siena Co-advisor: Dr. Mathilde Bouvel, University of Zurich A thesis submitted in fulfilment of the requirements for the degree of Doctor of Philosophy December 13, 2017 Abstract The study carried out along this dissertation fits into the field of enumerative combina- torics. The structures that have been investigated are families of discrete objects which display combinatorial properties. They are mainly subfamilies of well-known combinato- rial structures, such as lattice paths, pattern-avoiding permutations, polyominoes (more specifically, parallelogram polyominoes), or inversion sequences. Furthermore, these combi- natorial families are closely related to two famous number sequences known in the literature as the Catalan and Baxter numbers. We start defining and studying the properties of a combinatorial class whose elements appear as tilings of parallelogram polyominoes, and thus are called slicings of parallelogram polyominoes. This first combinatorial class results to be enumerated by Baxter numbers, and shows relations with Catalan and Schr¨odernumbers. Then, we turn to two families of pattern-avoiding permutations. First, we study the semi-Baxter permutations, which reveal a natural generalisation of the Baxter numbers. Next, we deal with the strong-Baxter permutations, which are defined by restricting the Baxter numbers. Thereafter, some generalisations of Catalan numbers are presented. A first generali- sation involves families of inversion sequences and lattice paths among the combinatorial structures enumerated, and is shown to be related to the sequence of Baxter numbers as well. Meanwhile, the family of fighting fish provide another generalisation of the Catalan numbers that appears to be independent from the Baxter numbers. These objects gen- eralise the family of parallelogram polyominoes, and display remarkable probabilistic and enumerative properties. In this dissertation we tackle the problem of enumerating these combinatorial classes and exhibiting their combinatorial properties, resolving some conjectures and open prob- lems. The methods used are rather diverse: for instance, establishing one-to-one correspon- dences with other structures, or combining the use of generating functions and succession rules. A succession rule is a powerful tool for counting discrete objects, and moreover it allows to generate them exhaustively. Owing to this remarkable fact succession rules, and equivalently generating trees, are largely used in our study of combinatorial structures. i ii Riassunto Il presente lavoro si inserisce nell'ambito della combinatoria, e pi`uprecisamente nel ramo della combinatoria enumerativa. L'oggetto di studio sono classi di oggetti discreti carat- terizzabili per mezzo di propriet`acombinatorie. In particolare, le strutture trattate sono sottofamiglie di note classi combinatorie quali i cammini nel piano, o le permutazioni a motivo escluso, o i poliomini parallelogrammi. Inoltre, le strutture studiate sono stretta- mente legate a due sequenze di numeri ben note in letteratura: i numeri di Catalan e i numeri di Baxter. La prima struttura combinatoria che abbiamo definito e di cui abbiamo studiato le pro- priet`a`euna particolare tassellatura dei poliomini parallelogrammi, che chiamiamo slicings of parallelogram polyominoes. Questa famiglia risulta essere contata dai numeri di Baxter, e legata ai numeri di Catalan e di Schr¨oder.Nel seguito presentiamo due famiglie di per- mutazioni a motivo escluso. Dapprima studiamo le semi-Baxter permutations, che rivelano una naturale generalizzazione dei numeri di Baxter; poi passiamo alle strong-Baxter per- mutations che si propongono, invece, come una loro naturale restrizione. Successivamente definiamo due diverse generalizzazioni dei numeri di Catalan. La prima generalizzazione coinvolge alcune famiglie di inversion sequences e di cammini nel piano, e presenta relazioni anche con i numeri di Baxter. La seconda, invece, si configura come una generalizzazione della famiglia dei poliomini parallelogrammi, e resta indipendente dai numeri di Baxter. Gli oggetti combinatori definiti nella seconda generalizzazione sono chiamati fighting fish e mostrano notevoli propriet`asia probabilistiche, che combinatorie. I problemi di enumerazione affrontati nella presente tesi dottorale utilizzano per la loro risoluzione diversi approcci. Ad esempio, alcuni risultati sono ottenuti stabilendo corrispon- denze biunivoche con altre strutture note, altri combinando l'uso di funzioni generatrici con quello di regole di successione. Le regole di successione sono un potente strumento enumerativo, che permette di generare esaustivamente gli oggetti di una data classe com- binatoria. Per tale motivo, gli alberi di generazione, e la loro formulazione come regole di successione, si configurano in questo lavoro come il principale strumento di enumerazione utilizzato. iii iv Ringraziamenti Giunta al termine di questi tre anni di dottorato, ho il desiderio di ringraziare tutti coloro che mi hanno accompagnato in questo percorso di ricerca scientifica e crescita personale, e senza i quali la realizzazione di questo lavoro non sarebbe stata possibile. Innanzitutto voglio ringraziare il mio tutor, Prof. Simone Rinaldi, per avermi fatto scoprire circa quattro anni fa questo ramo meno noto della matematica, chiamato combi- natoria. Lo ringrazio per avermi dato l'opportunit`adi fare ricerca in questo campo, e per essersi adoperato affinch´eio potessi fare esperienze costruttive e formative. Lo ringrazio, inoltre, per aver messo a mia disposizione il suo tempo e le sue conoscenze, per avermi donato i suoi preziosi consigli, e per avermi offerto il suo supporto nei momenti meno gratificanti che caratterizzano il lavoro di ricerca. Un altro speciale ringraziamento voglio rivolgerlo a Mathilde Bouvel, che come mia co-tutor, ha diretto assieme a Simone il mio programma di ricerca. La ringrazio per essersi sempre dimostrata disponibile anche nei periodi pi`uimpegnativi, e per essersi prodigata affich´epotessi intraprendere esperienze produttive e utili alla mia crescita come dottoranda di ricerca. La ringrazio per il lavoro che abbiamo svolto a Zurigo e a Siena, e per il suo contributo teorico e metodologico, nonch´eper la sua dedizione nel seguire le mie ricerche. La ringrazio, in particolare, per essere stata una persona comprensiva e gentile, e sulla quale fare completo affidamento. Voglio ringraziare, poi, sia Simone che Mathilde, perch´ehanno speso parte del loro tempo per leggere e discutere con me non solo le bozze di questo elaborato, ma anche gli altri progetti realizzati durante il dottorato. Ringrazio tutti coloro con i quali ho stabilito delle collaborazioni scientifiche, e i quali mi hanno arricchito trasferendomi parte delle loro conoscenze. In particolare, ringrazio Robert Cori, Nick R. Beaton, Andrew Rechnitzer, Enrica Duchi e Gilles Schaeffer. Devo un particolare ringraziamento ad Andrew per avermi gentilmente ospitato al Dipartimento di Matematica dell'Universit`adella Columbia Britannica, e per avermi mostrato l'utilizzo del metodo del kernel nella variante \obstinate". Ringrazio poi Enrica e Gilles che nei due anni trascorsi a Siena hanno condiviso con me non solo il loro sapere e l'amore per la ricerca, ma anche moltissimi piacevoli momenti di vita quotidiana. v Inoltre, voglio ringraziare tutti i colleghi e amici, che ho avuto il piacere di incontrare e di conoscere in questi anni. Con loro ho condiviso non solo le occasioni di lavoro e gli impegni del dottorato, ma anche le gradite pause caff´ee altri numerosi momenti che ricorder`osempre con il sorriso. Un ringraziamento speciale `erivolto ai miei amici pi`ucari, che specialmente nel periodo di stesura di questa tesi, mi hanno dimostrato il loro affetto supportandomi con pazienza. Voglio ringraziare, infine, ma in modo particolare, la mia famiglia per il loro sostegno e i loro consigli, per l'amore e la fiducia che mi sono stati dimostrati in ogni occasione. Veronica vi Contents Introduction 1 1 Introductory notions 7 1.1 Getting started . .7 1.1.1 Well-posed counting problems . .7 1.1.2 Lattice paths . .8 1.1.3 Pattern-avoiding permutations . .9 1.2 Methodology . 12 1.2.1 Bijective method . 12 1.2.2 ECO method . 13 1.2.3 Generating trees and succession rules . 14 1.2.4 Generating functions . 16 1.3 Introduction to Catalan structures . 21 1.3.1 Formulas . 21 1.3.2 Structures . 22 1.3.3 Bijections . 24 1.3.4 Generating trees . 28 1.3.5 Succession rules . 31 1.3.6 Catalan generating function: the kernel method . 33 1.3.7 Alternative use of the kernel method . 35 1.3.8 Asymptotics . 37 1.4 Introduction to Baxter structures . 37 1.4.1 Formulas . 37 1.4.2 Structures . 38 1.4.3 Bijections . 42 1.4.4 Generating trees and succession rules . 43 1.4.5 Baxter generating function: the obstinate variant of the kernel method ...................................... 47 1.4.6 Asymptotics . 51 2 Slicings of parallelogram polyominoes 53 2.1 Baxter slicings of parallelogram polyominoes . 54 2.1.1 Definition and growth of Baxter slicings . 54 vii viii CONTENTS 2.1.2 Bijection with triples of NILPs . 55 2.1.3 Definition and growth of Catalan slicings . 57 2.1.4 Definition of Baxter paths, and their bijection with Baxter slicings . 58 2.1.5 Baxter slicings of a given shape . 62 2.1.6 A discrete continuity . 62 2.2 Schr¨odernumbers . 63 2.2.1 Formulas and known structures . 63 2.2.2 Generating trees and succession rules . 67 2.3 A new Schr¨oderfamily of parking functions . 71 2.3.1 Schr¨oderparking functions . 72 2.3.2 Bijection between Schr¨oderparking functions and Schr¨oderpaths . 75 2.4 Schr¨oderslicings . 77 2.4.1 A new Schr¨odersuccession rule . 77 2.4.2 Definition of Schr¨oderslicings, and their growth . 80 2.5 Other Schr¨oderrestrictions of Baxter objects . 81 2.5.1 A Schr¨oderfamily of NILPs . 82 2.5.2 Another Schr¨odersubset of Baxter permutations . 83 2.5.3 A Schr¨oderfamily of mosaic floorplans . 85 2.6 Generalisation of Schr¨oderand Catalan slicings . 90 2.6.1 Skinny slicings . 90 2.6.2 Row-restricted slicings .
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