 Darij Grinberg and Victor
Reiner, [Prop] Hopf Algebras in
Combinatorics.
[Prop] Sourcecode of the notes, and
[Prop] a version with solutions to exercises.
The paper also appears as arXiv preprint arXiv:1409.8356, but the version on this website is updated more frequently.
These notes  originating from a onesemester class by Victor Reiner at the University of Minnesota  survey some of the most important Hopf algebras appearing in combinatorics. After introducing coalgebras, bialgebras and Hopf algebras in general, we study the Hopf algebra of symmetric functions, including Zelevinsky's axiomatic characterization of it as a "positive selfadjoint Hopf algebra" and its application to the representation theory of symmetric and (briefly) finite general linear groups. The notes then continue with the quasisymmetric and the noncommutative symmetric functions, some Hopf algebras formed from graphs, posets and matroids, and the MalvenutoReutenauer Hopf algebra of permutations. Among the results surveyed are the LittlewoodRichardson rule and other symmetric function identities, Zelevinsky's structure theorem for PSHs, the antipode formula for Ppartition enumerators, the AguiarBergeronSottile universal property of QSym, and Hazewinkel's polynomial freeness of QSym. The notes are written with a graduate student reader in mind, being mostly selfcontained but requiring a good familiarity with multilinear algebra and  for the representationtheory applications  basic group representation theory.
Comments are welcome (on anything here, but on these notes in particular)!
 λrings:
Definitions and basic properties.
These are notes I have made while learning this subject myself; they contain
just the basics of the theory (definitions of λrings and special
λrings, Adams operations, Todd homomorphisms and some more). I have
started them in 2011 chiefly to improve on the sloppiness of the λring
literature that I knew back then; as I am now aware, there are
much
better and more informative
references around.
It should be kept in mind that the notation I use is not the one prevalent in
modern literature. What Hazewinkel, in his
Witt vectors. Part 1, and Yau,
in his
LambdaRings,
call a "λring" is my "special λring", and what they call
"preλring" is my "λring". Also, Hazewinkel's Λ (A) is
slightly different
from mine (for example, where I set Π (K^{~}, [u_{1}, u_{2},
..., u_{n}]) = product (1 + u_{i}T) from 1 till n, he
sets Π (K^{~}, [u_{1}, u_{2}, ..., u_{n}])
= ∏ (1  u_{i}T)^{1} from 1 till n).

Darij Grinberg, Notes on the combinatorial
fundamentals of algebra.
PDF file.
Sourcecode and Github repository.
A version without solutions,
for spoilerless searching.
A set of notes on binomial coefficients, permutations and
determinants written for a
PRIMES
reading project 2015. 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).

Darij Grinberg, Notes on
linear algebra (work in progress).
PDF file.
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 (multiplications 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 Minneapolis.