Notes and papers on algebra and algebraic combinatorics
Darij Grinberg, Notes on the combinatorial
fundamentals of algebra.
Sourcecode and Github repository.
A version without solutions,
for spoilerless searching.
The paper also appears as arXiv preprint arXiv:2008.09862, but the version on this website is updated more frequently.
A set of notes on binomial coefficients, permutations and
determinants. Currently covers some binomial coefficient
identities (the Vandermonde convolution and some of its variations),
lengths and signs of permutations, and various elementary properties
of determinants (defined by the Leibniz formula).
The sourcecode of the project is also tracked
on github.
Darij Grinberg, Notes on
linear algebra (unfinished draft, currently frozen).
Github repository.
An attempt at a rigorous introduction to linear algebra, currently
frozen (as it has proven to be more work than I have time for). It
currently covers matrix operations (multiplication etc.), various
properties of matrices (triangularity, invertibility, permutation) and
some basics of vector spaces. It was written to accompany
my Math 4242 class
at the University of Minnesota.
Darij Grinberg, Enumerative Combinatorics
(unfinished draft, currently frozen).
Ancillary content (homeworks, sourcecode).
A rigorous introduction to enumerative combinatorics
at undergraduate level.
Currently, the first two chapters are finished,
covering binomial coefficients and basic counting strategies.
Darij Grinberg, Algebraic Combinatorics
(unfinished draft, currently frozen).
Sourcecode.
An introduction to algebraic combinatorics at early graduate
level. In its current stage of completion, it covers
the theory of formal power series in one variable; basic
properties of integer partitions up until the Jacobi Triple
Product; permutations; determinant identities; symmetric
polynomials up until the Littlewood-Richardson rule (proven
a la Stembridge).
Darij Grinberg, Mathematical Problem
Solving
(unfinished draft, currently frozen).
Sourcecode.
Notes on various parts of (mostly fairly elementary)
mathematics that appear in mathematical contests such
as the IMO and Putnam. Includes in-depth treatments of
elementary number theory, finite sums, the extremal
and pigeonhole principles, invariants, basic counting and
3-term linear recurrences.
Darij Grinberg, Alternierende Summen: Aufgaben
und Lösungen
(unfertig) (in German).
Sourcecode.
Eine Aufgabensammlung (mit teilweisen Lösungen)
über alternierende (d.h., vorzeichenbehaftete) Summen
in der Kombinatorik und (elementaren) Algebra.
Geschrieben für die deutsche IMO-Vorbereitung 2020.
Darij Grinberg,
Integrality over ideal
semifiltrations, preprint.
Detailed version with
expanded proofs.
Sourcecode of the paper.
Also available:
Algebra notes
Darij Grinberg