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On the rank of random subsets of finite affine geometry

Wojciech Kordecki — 2000

Discussiones Mathematicae Graph Theory

The aim of the paper is to give an effective formula for the calculation of the probability that a random subset of an affine geometry AG(r-1,q) has rank r. Tables for the probabilities are given for small ranks. The expected time to the first moment at which a random subset of an affine geometry achieves the rank r is derived.

Exact Expectation and Variance of Minimal Basis of Random Matroids

Wojciech KordeckiAnna Lyczkowska-Hanćkowiak — 2013

Discussiones Mathematicae Graph Theory

We formulate and prove a formula to compute the expected value of the minimal random basis of an arbitrary finite matroid whose elements are assigned weights which are independent and uniformly distributed on the interval [0, 1]. This method yields an exact formula in terms of the Tutte polynomial. We give a simple formula to find the minimal random basis of the projective geometry PG(r − 1, q).

Random matroids

CONTENTS1. Introduction.............................................................................52. Matroids..................................................................................6  2.1. Notations and basic properties...........................................6  2.2. Gaussian coefficients.......................................................10  2.3. Projective geometries.......................................................11  2.4. Special classes................................................................143....

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