Analytic Combinatorics is a self-contained treatment of the mathematics underlying the analysis of discrete structures, which has emerged over the past several decades as an essential tool in the understanding of properties of computer programs and scientific models with applications in physics, biology and chemistry. Thorough treatment of a large number of classical applications is an essential aspect of the presentation. Written by the leaders in the field of analytic combinatorics, this text is certain to become the definitive reference on the topic. The text is complemented with exercises, examples, appendices and notes to aid understanding therefore, it can be used as the basis for an advanced undergraduate or a graduate course on the subject, or for self-study.

Analytic Combinatorics

Groupes et algébres de lie

In One Line and Close. Permutations as Linear Orders. Descents Alternating Runs Alternating Subsequences In One Line and Anywhere. Permutations as Linear Orders. Inversions. Inversions Inversion in Permutations of Multisets In Many Circles. Permutations as Products of Cycles. Decomposing a Permutation into Cycles Type and Stirling Numbers Cycle Decomposition versus Linear Order Permutations with Restricted Cycle Structure In Any Way but This. Pattern Avoidance. The Basics. The Notion of Pattern Avoidance Patterns of Length Three Monotone Patterns Patterns of Length Four The Proof of the Stanley-Wilf Conjecture In This Way but Nicely. Pattern Avoidance. Follow-Up. Polynomial Recurrences Containing a Pattern Many Times Containing a Pattern a Given Number of Times Mean and Insensitive. Random Permutations. The Probabilistic Viewpoint Expectation Variance and Standard Deviation An Application: Longest Increasing Subsequences Permutations versus Everything Else. Algebraic Combinatorics of Permutations. The Robinson-Schensted-Knuth Correspondence Posets of Permutations Simplicial Complexes of Permutations Get Them All. Algorithms and Permutations. Generating Permutations Stack Sorting Permutations Variations of Stack Sorting How Did We Get Here? Permutations as Genome Rearrangements. Introduction Block Transpositions Block Interchanges Block Transpositions Revisited Solutions to Odd-Numbered Exercises References List of Frequently Used Notation Index Exercises, Problems, and Problem Solutions appear at the end of each chapter.

Combinatorics of Permutations

Combinatorics of permutations

The purpose of this article is threefold. First, it provides the reader with a few useful and efficient tools which should enable her/him to evaluate nontrivial determinants for the case such a determinant should appear in her/his research. Second, it lists a number of such determinants that have been already evaluated, together with explanations which tell in which contexts they have appeared. Third, it points out references where further such determinant evaluations can be found.

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Advanced determinant calculus

Processing blurred images is a key problem in many image applications. Existing methods to obtain blur invariants which are invariant with respect to centrally symmetric blur are based on geometric moments or complex moments. In this paper, we propose a new method to construct a set of blur invariants using the orthogonal Legendre moments. Some important properties of Legendre moments for the blurred image are presented and proved. The performance of the proposed descriptors is evaluated with various point-spread functions and different image noises. The comparison of the present approach with previous methods in terms of pattern recognition accuracy is also provided. The experimental results show that the proposed descriptors are more robust to noise and have better discriminative power than the methods based on geometric or complex moments.

https://www.hal.inserm.fr/inserm-00437044/document

Blurred Image Recognition by Legendre Moment Invariants

Nekrasov et Okounkov ont obtenu une nouvelle formule pour le developpement des puissances du produit d'Euler, a l'aide des longueurs d'equerre des partitions d'entiers, dans leur etude de la theorie de Seiberg-Witten. Nous proposons un raffinement de cette formule reposant sur une propriete nouvelle des t-cores, qui permet de donner une demonstration elementaire en faisant usage des identites de Macdonald. Nous obtenons aussi une extension, en ajoutant deux parametres supplementaires, qui peut etre consideree comme une interpolation discrete entre les identites de Macdonald et la fonction generatrice des t-cores. Plusieurs applications en sont deduites, y compris la "formule d'equerre pointee".

/pdf/the-nekrasov-okounkov-hook-length-formula-refinement-9j80v63rev.pdf

The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications

This paper proposes a novel fractional transform, denoted as the fractional Krawtchouk transform (FrKT), a generalization of the Krawtchouk transform. The derivation of the FrKT uses the eigenvalue decomposition method. We determine the eigenvalues and the corresponding multiplicity of the Krawtchouk transform matrix. Moreover, the orthonormal eigenvectors of the transform matrix are derived. For validation purpose only and as a first illustration of the interest of FrKT, a watermarking example was chosen. Experimental results show that better watermark robustness and imperceptibility are achieved by adjusting the fractional orders in the FrKT.

Fractional Krawtchouk Transform With an Application to Image Watermarking

https://www.hal.inserm.fr/inserm-00421043/document

Guo-Niu Han

Papers

Blurred Image Recognition by Legendre Moment Invariants

The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications

Fractional Krawtchouk Transform With an Application to Image Watermarking

Reconstruction of tomographic images from limited range projections using discrete Radon transform and Tchebichef moments

Efficient Legendre moment computation for grey level images