Enumerating five families of pattern-avoiding inversion sequences; and introducing the powered Catalan numbers

The first problem addressed by this article is the enumeration of some families of pattern-avoiding inversion sequences. We solve some enumerative conjectures left open by the foundational work on the topics by Corteel et al., some of these being also solved independently by Lin, and Kim and Lin. The strength of our approach is its robustness: we enumerate four families $F_1 \subset F_2 \subset F_3 \subset F_4$ of pattern-avoiding inversion sequences ordered by inclusion using the same approach. More precisely, we provide a generating tree (with associated succession rule) for each family $F_i$ which generalizes the one for the family $F_{i-1}$. The second topic of the paper is the enumeration of a fifth family $F_5$ of pattern-avoiding inversion sequences (containing $F_4$). This enumeration is also solved \emph{via} a succession rule, which however does not generalize the one for $F_4$. The associated enumeration sequence, which we call the \emph{powered Catalan numbers}, is quite intriguing, and further investigated. We provide two different succession rules for it, denoted $\Omega_{pCat}$ and $\Omega_{steady}$, and show that they define two types of families enumerated by powered Catalan numbers. Among such families, we introduce the \emph{steady paths}, which are naturally associated with $\Omega_{steady}$. They allow us to bridge the gap between the two types of families enumerated by powered Catalan numbers: indeed, we provide a size-preserving bijection between steady paths and valley-marked Dyck paths (which are naturally associated with $\Omega_{pCat}$). Along the way, we provide several nice connections to families of permutations defined by the avoidance of vincular patterns, and some enumerative conjectures.Comment: V2 includes modifications suggested by referees (in particular, a much shorter Section 3, to account for arXiv:1706.07213

An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences

We introduce an algorithmic approach based on generating tree method for enumerating the inversion sequences with various pattern-avoidance restrictions. For a given set of patterns, we propose an algorithm that outputs either an accurate description of the succession rules of the corresponding generating tree or an ansatz. By using this approach, we determine the generating trees for the pattern-classes $I_n(000, 021), I_n(100, 021)$, $I_n(110, 021), I_n(102, 021)$, $I_n(100,012)$, $I_n(011,201)$, $I_n(011,210)$ and $I_n(120,210)$. Then we use the kernel method, obtain generating functions of each class, and find enumerating formulas. Lin and Yan studied the classification of the Wilf-equivalences for inversion sequences avoiding pairs of length-three patterns and showed that there are 48 Wilf classes among 78 pairs. In this paper, we solve six open cases for such pattern classes.Comment: 20 pages, 2 figure

Slicings of parallelogram polyominoes: Catalan, schröder, baxter, and other sequences

We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the m-skinny slicings and the m-rowrestricted slicings, for m ∈ N. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any m

Information theoretic refinement criteria for image synthesis

Aquest treball està enmarcat en el context de gràfics per computador partint de la intersecció de tres camps: rendering, teoria de la informació, i complexitat.Inicialment, el concepte de complexitat d'una escena es analitzat considerant tres perspectives des d'un punt de vista de la visibilitat geomètrica: complexitat en un punt interior, complexitat d'una animació, i complexitat d'una regió. L'enfoc principal d'aquesta tesi és l'exploració i desenvolupament de nous criteris de refinament pel problema de la il·luminació global. Mesures de la teoria de la informació basades en la entropia de Shannon i en la entropia generalitzada de Harvda-Charvát-Tsallis, conjuntament amb les f-divergències, són analitzades com a nuclis del refinement. Mostrem com ens aporten una rica varietat d'eficients i altament discriminatòries mesures que són aplicables al rendering en els seus enfocs de pixel-driven (ray-tracing) i object-space (radiositat jeràrquica).Primerament, basat en la entropia de Shannon, es defineixen un conjunt de mesures de qualitat i contrast del pixel. S'apliquen al supersampling en ray-tracing com a criteris de refinement, obtenint un algorisme nou de sampleig adaptatiu basat en entropia, amb un alt rati de qualitat versus cost. En segon lloc, basat en la entropia generalitzada de Harvda-Charvát-Tsallis, i en la informació mutua generalitzada, es defineixen tres nous criteris de refinament per la radiositat jeràrquica. En correspondencia amb tres enfocs clàssics, es presenten els oracles basats en la informació transportada, el suavitzat de la informació, i la informació mutua, amb resultats molt significatius per aquest darrer. Finalment, tres membres de la familia de les f-divergències de Csiszár's (divergències de Kullback-Leibler, chi-square, and Hellinger) son analitzats com a criteris de refinament mostrant bons resultats tant pel ray-tracing com per la radiositat jeràrquica.This work is framed within the context of computer graphics starting out from the intersection of three fields: rendering, information theory, and complexity.Initially, the concept of scene complexity is analysed considering three perspectives from a geometric visibility point of view: complexity at an interior point, complexity of an animation, and complexity of a region. The main focus of this dissertation is the exploration and development of new refinement criteria for the global illumination problem. Information-theoretic measures based on Shannon entropy and Harvda-Charvát-Tsallis generalised entropy, together with f-divergences, are analysed as kernels of refinement. We show how they give us a rich variety of efficient and highly discriminative measures which are applicable to rendering in its pixel-driven (ray-tracing) and object-space (hierarchical radiosity) approaches.Firstly, based on Shannon entropy, a set of pixel quality and pixel contrast measures are defined. They are applied to supersampling in ray-tracing as refinement criteria, obtaining a new entropy-based adaptive sampling algorithm with a high rate quality versus cost. Secondly, based on Harvda-Charvát-Tsallis generalised entropy, and generalised mutual information, three new refinement criteria are defined for hierarchical radiosity. In correspondence with three classic approaches, oracles based on transported information, information smoothness, and mutual information are presented, with very significant results for the latter. And finally, three members of the family of Csiszár's f-divergences (Kullback-Leibler, chi-square, and Hellinger divergences) are analysed as refinement criteria showing good results for both ray-tracing and hierarchical radiosity