27 research outputs found
321-polygon-avoiding permutations and Chebyshev polynomials
A 321-k-gon-avoiding permutation pi avoids 321 and the following four
patterns: k(k+2)(k+3)...(2k-1)1(2k)23...(k+1),
k(k+2)(k+3)...(2k-1)(2k)123...(k+1), (k+1)(k+2)(k+3)...(2k-1)1(2k)23...k,
(k+1)(k+2)(k+3)...(2k-1)(2k)123...k. The 321-4-gon-avoiding permutations were
introduced and studied by Billey and Warrington [BW] as a class of elements of
the symmetric group whose Kazhdan-Lusztig, Poincare polynomials, and the
singular loci of whose Schubert varieties have fairly simple formulas and
descriptions. Stankova and West [SW] gave an exact enumeration in terms of
linear recurrences with constant coefficients for the cases k=2,3,4. In this
paper, we extend these results by finding an explicit expression for the
generating function for the number of 321-k-gon-avoiding permutations on n
letters. The generating function is expressed via Chebyshev polynomials of the
second kind.Comment: 11 pages, 1 figur
Enumeration schemes for restricted permutations
Zeilberger's enumeration schemes can be used to completely automate the
enumeration of many permutation classes. We extend his enumeration schemes so
that they apply to many more permutation classes and describe the Maple package
WILFPLUS, which implements this process. We also compare enumeration schemes to
three other systematic enumeration techniques: generating trees, substitution
decompositions, and the insertion encoding.Comment: 21 page
Investigation into the Accuracy and Practicality of Methods for Transforming Coordinates between Geodetic Datums
This thesis is a study of methods of transforming coordinates between geodetic datums, the methods being generally known as datum transformations.
Direct methods are described and categorised as conformal, near-conformal and non-conformal. New variations on all three types are included in the direct methods: SMITSWAM (which avoids changes of coordinate-type), generalisations of Standard & Abridged Molodensky, and normalised generalisations of multiple regression equations (5 types). Reverse transformations are extensively covered, as are methods of derivation. In both cases, new algorithms are included.
Direct methods, with the exception of multiple regression equations, do not capture distortions in datum transformations. The thesis therefore includes a review of composite methods which extract a trend model and apply a surface-fitting technique (SFT) to the residuals. Sometimes the SFT is used as a gridding method, producing regularly-spaced data that can be interpolated as a final stage of the composite process.
The SFTs selected for detailed study include new variations on inverse-distance-to-a-power weighting and nearest-neighbour interpolation. These are called HIPFEAD and LIVONN respectively. In both cases, the variations are shown to have advantages in terms of accuracy of fit. Least-squares collocation and radial basis functions are shown to produce reusable vectors - described here as “revamped signals” – that enable interpolation without gridding.
Where the composite methods are used for gridding, it is shown that geodetic coordinates can be used, avoiding the need for projected grid coordinates. The interpolation options applied are piecewise-bilinear and piecewise-bicubic, the latter being an algorithm (believed to be new) that uses up to 12 “grid” points.
Case studies were considered using 6 datasets, two for Great Britain, one each for Western Australia, Ghana, Sweden and Slovenia. These showed beneficial properties of the new methods, both in the direct and composite categories. They also enabled comparisons of transformation methods generally
Model-Based Problem Solving through Symbolic Regression via Pareto Genetic Programming.
Pareto genetic programming methodology is extended by additional generic model selection and generation strategies that (1) drive the modeling engine to creation of models of reduced non-linearity and increased generalization capabilities, and (2) improve the effectiveness of the search for robust models by goal softening and adaptive fitness evaluations. In addition to the new strategies for model development and model selection, this dissertation presents a new approach for analysis, ranking, and compression of given multi-dimensional input-response data for the purpose of balancing the information content of undesigned data sets.
Analytic Combinatorics in Several Variables: Effective Asymptotics and Lattice Path Enumeration
The field of analytic combinatorics, which studies the asymptotic behaviour
of sequences through analytic properties of their generating functions, has led
to the development of deep and powerful tools with applications across
mathematics and the natural sciences. In addition to the now classical
univariate theory, recent work in the study of analytic combinatorics in
several variables (ACSV) has shown how to derive asymptotics for the
coefficients of certain D-finite functions represented by diagonals of
multivariate rational functions. We give a pedagogical introduction to the
methods of ACSV from a computer algebra viewpoint, developing rigorous
algorithms and giving the first complexity results in this area under
conditions which are broadly satisfied. Furthermore, we give several new
applications of ACSV to the enumeration of lattice walks restricted to certain
regions. In addition to proving several open conjectures on the asymptotics of
such walks, a detailed study of lattice walk models with weighted steps is
undertaken.Comment: PhD thesis, University of Waterloo and ENS Lyon - 259 page
LIPIcs, Volume 248, ISAAC 2022, Complete Volume
LIPIcs, Volume 248, ISAAC 2022, Complete Volum
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Large bichromatic point sets admit empty monochromatic 4-gons
We consider a variation of a problem stated by ErdËťos
and Szekeres in 1935 about the existence of a number
fES(k) such that any set S of at least fES(k) points in
general position in the plane has a subset of k points
that are the vertices of a convex k-gon. In our setting
the points of S are colored, and we say that a (not necessarily
convex) spanned polygon is monochromatic if
all its vertices have the same color. Moreover, a polygon
is called empty if it does not contain any points of
S in its interior. We show that any bichromatic set of
n ≥ 5044 points in R2 in general position determines
at least one empty, monochromatic quadrilateral (and
thus linearly many).Postprint (published version