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

https://www-lipn.univ-paris13.fr/~banderier/Papers/tcs_banderier_flajolet_2002.pdf

Basic analytic combinatorics of directed lattice paths

Digital geometry is the study of geometrical properties of subsets of digital images. If the digitization is sufficiently fine-grained, such properties can be regarded as approximations to the corresponding properties of the "real" sets that gave rise, by digitization, to the digital sets; but it is also important to define how the properties can be computed for the digital sets themselves. Questions of particular interest include how images and image subsets are digitized; how geometric properties are defined for digitized sets; the computational complexity of computing them--in particular, whether they can be computed using simple (e.g., local) operations; characterizing image operations that preserve them; and characterizing digital objects that could be the digitizations of real objects that have given geometric properties. Concepts that have been extensively studied include topological properties (connected components, boundaries); curves and surfaces; straightness, curvature, convexity, and elongatedness; distance, extent, length, area, surface area, volume, and moments; shape description, similarity, symmetry, and relative position; shape simplification and skeletonization.

Digital geometry

The worst case fall time is given by tf = 2.2 (2Rn•Cout + Rn•Cx) = 2.2(2*1 KΩ*8fF + 1KΩ*2fF) = 39.6ps b) The best-case rise time for a NAND gate occurs when both pMOS turn on and is given by tr = 2.2 (Rp/2) •Cout, where the total parallel resistance is Rp/2. = 2.2 * 2 KΩ/2 * 8 fF = 17.6 ps c) The worst-case rise time of a NOR gate is given by tr = 2.2 (2Rp•Cout + Rp•Cx) = 2.2(2*2 KΩ*8fF + 2KΩ*2fF) = 79.2 ps

Problem (2)

Completing the genome sequence of an organism is an important task in comparative, functional and structural genomics. However, this remains a challenging issue from both a computational and an experimental viewpoint. Genome scaffolding (i.e. the process of ordering and orientating contigs) of de novo assemblies usually represents the first step in most genome finishing pipelines. In this paper, we present MeDuSa (Multi-Draft based Scaffolder), an algorithm for genome scaffolding. MeDuSa exploits information obtained from a set of (draft or closed) genomes from related organisms to determine the correct order and orientation of the contigs. MeDuSa formalises the scaffolding problem by means of a combinatorial optimisation formulation on graphs and implements an efficient constant factor approximation algorithm to solve it. In contrast to currently used scaffolders, it does not require either prior knowledge on the microrganisms dataset under analysis (e.g. their phylogenetic relationships) or the availability of paired end read libraries. This makes usability and running time two additional important features of our method. Moreover, benchmarks and tests on real bacterial datasets showed that MeDuSa is highly accurate and, in most cases, outperforms traditional scaffolders. The possibility to use MeDuSa on eukaryotic datasets has also been evaluated, leading to interesting results. MeDuSa web server: http://combo.dbe.unifi.it/medusa A stand-alone version of the software can be downloaded from https://github.com/combogenomics/medusa/releases. All results presented in this work have been obtained with MeDuSa v. 1.3. marco.fondi@unifi.it.

MeDuSa: a multi-draft based scaffolder

Reconstruction of 4- and 8-connected convex discrete sets from row and column projections

https://hal.archives-ouvertes.fr/docs/00/06/61/77/PDF/tomoqconv_els.pdf

Sara Brunetti

Papers

MeDuSa: a multi-draft based scaffolder

Reconstruction of 4- and 8-connected convex discrete sets from row and column projections

An algorithm reconstructing convex lattice sets

Discrete tomography determination of bounded lattice sets from four X-rays

Stability results for the reconstruction of binary pictures from two projections