Cette thèse se divise en deux grandes parties, la première traite la combinatoire associée à l’ordre normal des opérateurs non-commutatifs et la seconde aborde des distributions symétriques du nombre de croisements et du nombre d’emboîtements, respectivement k-croisements et k-emboîtements, dans des structures combinatoires (partitions, permutations, permutations colorées, …). La première partie étudie l’ordre normal des opérateurs en termes de placements de tours. Nous étudions la forme de l’ordre normal en connectant deux opérateurs non-commutatifs D et U, et des polynômes orthogonaux spéciaux, et établissons des bijonctions entre les coefficients de (D+U)n et le nombre de placements de tours sur un diagramme de Ferrers. Nous donnons également des preuves combinatoires à des conjectures quantiques posées par des physiciens. Dans la seconde partie, nous définissons des statistiques, comme emboîtements et k-emboîtements, sur l’ensemble des permutations du groupe de Coxeter de type B. Nous donnons également des extensions au type B des résultats sur les croisements et les emboîtements, respectivement k-croisements et k-emboîtements dans les permutations de type A, en termes de distributions symétriques. De plus, nous étudions le lien entre les opérateurs non-commutatifs et ces statistiques. D’autres extensions de la distribution de ces statistiques sur les ensembles de partitions colorées et de permutations colorées de types A et B sont ainsi établies / This thesis is divided into two parts, the first deals with the combinatorics associated to the normal ordering form of noncommutative operators and the second addresses the symmetric distributions of the crossing numbers and nesting numbers, respectively k-crossings and k-nestings, in combinatorial structures (partitions, permutations, colored permutations, …). The first part studies the normal order of operators in terms of rook placements. We study the normal ordering form connecting two noncommutative operators D and U, and some special orthogonal polynomials, and establish bijonctions between coefficients of (D+U)n and rook placements in Ferrers diagrams. We also give combinatorial proofs and alternatives to some quantum conjectures posed by physicists. In the second part, we define the notions of statistics, nestings and k-nestings, on the sets of permutations of the Coxeter group of type B. We also give extensions to type B of the results of the crossings and nestings, respectivelu k-crossings and K-nestings in the set of permutations of type A, in terms of symmetric distributions. Likewise, we study the link between non-commutative operators and these statistics. Other extensions of the distribution of these statistics on the sets of colored partitions and colored permutations of type A and B are established

<p>The computer vision task of classifying natural images is a primary driving force behind modern AI algorithms. Deep Convolutional Neural Networks (CNNs) demonstrate state of the art performance in large scale multi-class image classification tasks. However, due to the many layers and millions of parameters these models are considered to be black box algorithms. The decisions of these models are further obscured due to a cumbersome multi-class decision process. There exists another approach called class binarization in the literature which determines the multi-class prediction outcome through a sequence of binary decisions.The focus of this dissertation is on the integration of the class-binarization approach to multi-class classification with deep learning models, such as CNNs, for addressing large scale image classification problems. Three works are presented to address the integration.</p> <p>In the first work, Error Correcting Output Codes (ECOCs) are integrated into CNNs by inserting a latent-binarization layer prior to the CNNs final classification layer.  This approach encapsulates both encoding and decoding steps of ECOC into a single CNN architecture. EM and Gibbs sampling algorithms are combined with back-propagation to train CNN models with Latent Binarization (LB-CNN). The training process of LB-CNN guides the model to discover hidden relationships similar to the semantic relationships known apriori between the categories. The proposed models and algorithms are applied to several image recognition tasks, producing excellent results.</p> <p>In the second work, Hierarchically Decodeable Output Codes (HD-OCs) are proposedto compactly describe a hierarchical probabilistic binary decision process model over the features of a CNN. HD-OCs enforce more homogeneous assignments of the categories to the dichotomy labels. A novel concept called average decision depth is presented to quantify the average number of binary questions needed to classify an input. An HD-OC is trained using a hierarchical log-likelihood loss that is empirically shown to orient the output of the latent feature space to resemble the hierarchical structure described by the HD-OC. Experiments are conducted at several different scales of category labels. The experiments demonstrate strong performance and powerful insights into the decision process of the model.</p> <p>In the final work, the literature of enumerative combinatorics and partially ordered sets isused to establish a unifying framework of class-binarization methods under the Multivariate Bernoulli family of models. The unifying framework theoretically establishes simple relationships for transitioning between the different binarization approaches. Such relationships provide useful investigative tools for the discovery of statistical dependencies between large groups of categories. They are additionally useful for incorporating taxonomic information as well as enforcing structural model constraints. The unifying framework lays the groundwork for future theoretical and methodological work in addressing the fundamental issues of large scale multi-class classification.</p> <p><br></p>

Cette thèse regroupe plusieurs travaux de combinatoire énumérative sur les permutations et permutations d'ensemble. Elle comporte 4 parties.Dans la première partie, nous répondons aux conjectures de Steingrimsson sur les partitions ordonnées d'ensemble. Plus précisément, nous montrons que les statistiques de Steingrimsson sur les partitions ordonnées d'ensemble ont la distribution euler-mahonienne. Dans la deuxième partie, nous introduisons et étudions une nouvelle classe de statistiques sur les mots : les statistiques "maj-inv". Ces dernières sont des interpolations graphiques des célèbres statistiques "indice majeur" et "nombre d'inversions". Dans la troisième partie, nous montrons que la distribution conjointe des statistiques"nombre de croisements" et "nombre d'imbrications" sur les partitions d'ensemble est symétrique. Nous étendrons aussi ce dernier résultat dans le cadre beaucoup plus large des 01-remplissages de "polyominoes lunaires".La quatrième et dernière partie est consacrée à l'étude combinatoire des q-polynômes de Laguerre d'Al-Salam-Chihara. Nous donnerons une interprétation combinatoire de la suite de moments et des coefficients de linéarisations de ces polynômes. / This thesis consists of four chapters, each on a different topic in enumerative combinatorics, all related in some way to the enumeration of permutations or set partitions. In the first chapter, we prove and generalize Steingrimsson's conjectures on Euler-Mahonian statistics on ordered set partitions. In the second chapter, we introduce and study a new class of statistics on words: the "maj-inv" statistics. These are graphical interpolation of the well-known "major index" and "inversion number".In the third chapter, we show that the joint distribution of the numbers of crossings and nestings on set partitions is symmetric. We also put this result in the larger context of enumeration of increasing and decreasing chains in 01-fillings of moon polyominoes.In the last chapter, we decribe various aspects of the Al-Salam-Chihara q-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients.

We establish the existence of a symmetry within the Gromov-Witten theory of CPn and its blowup along points. The nature of this symmetry is encoded in the Cremona transform and its resolution, which lives on the toric variety of the permutohedron. This symmetry expresses some difficult to compute invariants in terms of others less difficult to compute. We focus on enumerative implications; in particular this technique yields a one line proof of the uniqueness of the rational normal curve. Our method involves a study of the toric geometry of the permutohedron, and degeneration of Gromov-Witten invariants. / En este trabajo establecemos la existencia de una simetra en el marco de la teora de Gromov-Witten para CPn y su explosion a lo largo de puntos. La naturaleza de esta simetra queda codicada en la transformacion de Cremona y su resolucion en una variedad torica del permutoedro. Esta simetra expresa algunos invariantes difciles de calcular junto con otros que no lo son tanto. Nos centramos en implicaciones enumerativas; en particular esta tecnica ofrece una prueba enuna lnea de la unicidad de la curva racional normal. Nuestro metodo involucra un estudio de la geometra torica del permutoedro, as como el de la degeneracion de los invariantes de Gromov-Witten.