This book is mostly based on the author's 25 years of teaching
combinatorics to two distinct sets of students: first-year students
and seniors from all backgrounds, not just limited to only those
majoring in mathematics and physics. The prerequisites are kept to
a minimum; essentially, only high school algebra is required. The
design is to go from zero knowledge to advanced themes and various
applications during a semester of three or three and a half months
with quite a few topics intended for research projects and
additional reading.This unique book features the key themes of
classical introductory combinatorics, modeling (mainly linear), and
elementary number theory with a constant focus on applications in
statistics, physics, biology, economics, and computer science.
These applications include dimers, random walks, binomial and
Poisson distributions, games of chance (lottery, dice, poker,
roulette), pricing options, population growth, tree growth,
modeling epidemic spread, invasion ecology, fission reactors, and
networks.A lot of material is provided in the form of relatively
self-contained problems, about 135, and exercises, about 270, which
are almost always with hints and answers. A systematic introduction
to number theory (with complete justifications) is a significant
part of the book, including finite fields, Pell's equations,
continued fractions, quadratic reciprocity, the Frobenius coin
problem, Pisano periods, applications to magic and Latin squares
and elements of cryptography. The recurrence relations and modeling
play a very significant role, including the usage of Bessel
functions for motivated readers. The book contains a lot of history
of mathematics and recreational mathematics.
General
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