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Secret history : the story of cryptology
\"Codes are a part of everyday life, from the ubiquitous Universal Price Code (UPC) to postal zip codes. They need not be intended for secrecy. They generally use groups of letters (sometimes pronounceable code words) or numbers to represent other words or phrases. There is typically no mathematical rule to pair an item with its representation in code. A few more examples will serve to illustrate the range of codes\"-- Provided by publisher.
Book
Decorated Dyck Paths, Polyominoes, and the Delta Conjecture
We discuss the combinatorics of decorated Dyck paths and decorated parallelogram polyominoes, extending to the decorated case the
main results of both Haglund (“A proof of the
eBook
Fixed Point Homing Shuffles
We study a family of maps from$S_n \\to S_n$we call fixed point homing shuffles. These maps generalize a few known problems such as Conway's Topswops, and a card shuffling process studied by Gweneth McKinley. We show that the iterates of these homing shuffles always converge, and characterize the set$U_n$of permutations that no homing shuffle sorts. We also study a homing shuffle that sorts anything not in$U_n$ , and find how many iterations it takes to converge in the worst case. Updated formatting to fit with DMTCS requirements
Journal Article
The combinatorics of Farey words and their traces
We introduce a family of 3-variable ‘Farey polynomials’ that are closely connected with the geometry and topology of 3-manifolds and orbifolds as they can be used to produce concrete realisations of the boundaries and local coordinates for one-dimensional (over C) deformation spaces of Kleinian groups. As such, this family of polynomials has a number of quite remarkable properties. We study these polynomials from an abstract combinatorial viewpoint, including a recursive definition extending that which is known in the literature for the special case of manifolds, even beyond what the geometry predicts. We also present some intriguing examples and conjectures which we would like to bring to the attention of researchers interested in algebraic combinatorics and hypergeometric functions. The results in this paper additionally provide a practical approach to various classification problems for rank 2 subgroups of PSL(2,C) since they, together with other recent work of the authors, make it possible to provide certificates that certain groups are discrete and free, and effective ways to identify relators.
Journal Article
Weight Multiplicities and Young Tableaux Through Affine Crystals
The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is
hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights of affine
Kac–Moody algebras are not known in most cases. In this paper, we study weight multiplicities for both finite and affine cases of
classical types for certain infinite families of highest weights modules. We introduce new classes of Young tableaux, called the
eBook
Generalized Triangular Numbers and Combinatorial Explanations
The formula for the sums of the first integers, which are known as triangular numbers, is well known and there are many proofs for it: by induction, graphical, by combinatorics, etc. The sum of the first triangular numbers is known as tetrahedral numbers. In this article1, we discuss a generalization of triangular and tetrahedral numbers where the number of summation symbols is variable. We repeat results from the literature that state that these so-called generalized triangular numbers can be represented via multicombinations, i.e. combinations with repetitions, and give an illustrative explanation for this formula, which is based on combinatorics. Via high-dimensional illustrations, we show that these generalized triangular numbers are figurate numbers, namely hyper-tetrahedral numbers, see Figure 1. Additionally, we demonstrate that there is a relation between the height and the dimension of these hypertetrahedra, i.e. a series of generalized triangular numbers with fixed dimension and varying height can be represented as such a series with fixed height and varying dimension, and vice versa.
Journal Article
Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry
In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements,
interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the
monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left
regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such
structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left
regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order
complexes of posets naturally associated to the left regular band.
The purpose of the present monograph is to further develop and
deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all
simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left
regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the
examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure
on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional
oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A
fairly complete picture of the representation theory for CW left regular bands is obtained.
eBook
Bounded Littlewood identities
We describe a method, based on the theory of Macdonald–Koornwinder polynomials, for proving bounded Littlewood identities. Our
approach provides an alternative to Macdonald’s partial fraction technique and results in the first examples of bounded Littlewood
identities for Macdonald polynomials. These identities, which take the form of decomposition formulas for Macdonald polynomials of type
eBook
A statistical model of COVID-19 testing in populations
We develop a statistical model for the testing of disease prevalence in a population. The model assumes a binary test result, positive or negative, but allows for biases in sample selection and both type I (false positive) and type II (false negative) testing errors. Our model also incorporates multiple test types and is able to distinguish between retesting and exclusion after testing. Our quantitative framework allows us to directly interpret testing results as a function of errors and biases. By applying our testing model to COVID-19 testing data and actual case data from specific jurisdictions, we are able to estimate and provide uncertainty quantification of indices that are crucial in a pandemic, such as disease prevalence and fatality ratios.
This article is part of the theme issue ‘Data science approach to infectious disease surveillance’.
Journal Article