Notes on the combinatorial fundamentals of algebra

Abstract

This is a detailed survey -- with rigorous and self-contained proofs -- of some of the basics of elementary combinatorics and algebra, including the properties of finite sums, binomial coefficients, permutations and determinants. It is entirely expository (and written to a large extent as a repository for folklore proofs); no new results (and few, if any, new proofs) appear.

1403 pages. v3 corrects typos and adds Exercises 6.65--6.69. Not a textbook; rather a repository of proofs I could cite. Posted here for easier referencing (and long-term archival). This project is tracked on https://github.com/darijgr/detnotes

References in corpus (2)

• Chio Condensation and Random Sign Matrices
• The common-submatrix Laplace expansion

Cited by in corpus (5)

• The Lehmer factorial norm on $S_n$
• The Schur--Weyl graph and Thoma's theorem
• The pre-Pieri rules
• Three variations on the linear independence of grouplikes in a coalgebra
• The Bhargava greedoid as a Gaussian elimination greedoid