• Corpus ID: 124485523

Notes on the combinatorial fundamentals of algebra

  title={Notes on the combinatorial fundamentals of algebra},
  author={Darij Grinberg},
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. 

The 4-periodic spiral determinant

The purpose of this note is to generalize the determinant formula conjectured by Amdeberhan in [Amdebe17] and outline how it can be proven. (Unfortunately, neither the generalization nor its proof

The pre-Pieri rules

Let R be a commutative ring and n ≥ 1 and p ≥ 0 two integers. Let N = {0, 1, 2, . . .}. Let hk, i be an element of R for all k ∈ Z and i ∈ [n]. For any α ∈ Zn, we define

Compound matrices in systems and control theory: a tutorial

The multiplicative and additive compounds of a matrix play an important role in several fields of mathematics including geometry, multi-linear algebra, combinatorics, and the analysis of nonlinear

Compound matrices in systems and control theory

The multiplicative and additive compounds of a matrix play an important role in geometry, multi-linear algebra, combinatorics , and the analysis of nonlinear time-varying dynamical systems. There is

The Schur–Weyl Graph and Thoma’s Theorem

We define a graded graph, called the Schur–Weyl graph, which arises naturally when one considers simultaneously the RSK algorithm and the classical duality between representations of the symmetric

On the principal minors of the powers of a matrix

. We show that if A is an n × n -matrix, then the diagonal entries of each power A m are uniquely determined by the principal minors of A , and can be written as universal (integral) polynomials in

Three variations on the linear independence of grouplikes in a coalgebra

The grouplike elements of a coalgebra over a field are known to be linearly independent over said field. Here we prove three variants of this result. One is a generalization to coalgebras over a

Combinatorial proof of Chio Pivotal Condensation

We refer to [Grinbe15] for undefined notations used here (though they should all be standard). Classically, Theorem 0.1 is proven using a trick. Namely, it is first proven under the assumption that

An exercise on determinant-like sums

Exercise 1. Let N = {0, 1, 2, . . .}. Let n ∈ N and r ∈ N with r > 0. The sum of a matrix shall mean the sum of its entries. If A is any matrix and i and j are two positive integers, then the (i,

) : homework set 4 with solutions 0 . 1 . On the Euler totient function

Let us first recall some basic facts from elementary number theory. A common divisor of two integers a and b is an integer that divides both a and b. If a and b are two integers satisfying (a, b) 6=



An Introduction to Abstract Algebra

This chapter describes a variety of basic algebraic structures that play roles in the generation and analysis of sequences, especially sequences intended for use in communications and cryptography.

A Course in Enumeration

Basics.- Fundamental Coefficients.- Formal Series and Infinite Matrices.- Methods.- Generating Functions.- Hypergeometric Summation.- Sieve Methods.- Enumeration of Patterns.- Topics.- The Catalan

On the Solution Space of Homogeneous Linear Recurrence Relations with Constant Coefficients

In this papre,using some basic knowledge of higher algebra,we discuss the algebraic structure of the solution set of homogeneous liner recurrence relations with constant coefficients.In fact,we prove

The Laurent Phenomenon

A unified treatment of the phenomenon of birational maps given by Laurent polynomials is suggested, which covers a large class of applications and settles in the affirmative a conjecture of D. Gale on integrality of generalized Somos sequences.


In this paper, the authors establish some identities involving inverses of binomial coefficients and generalize an identity.

A Path to Combinatorics for Undergraduates: Counting Strategies

Preface.- Introduction.- Acknowledgments.- Abbreviations and Notations.- Addition on Multiplication?- Combinations.- Properties of Binomial Coefficients.- Bijections.- Inclusions and Exclusions.-

Hopf Algebras in Combinatorics

These notes -- originating from a one-semester class by their second author at the University of Minnesota -- survey some of the most important Hopf algebras appearing in combinatorics. After

Proofs that Really Count: The Art of Combinatorial Proof

1. Fibonacci identities 2. Lucas identities 3. Gibonacci identities 4. Linear recurrences 5. Continued fractions 6. Binomial identities 7. Alternating sign binomial identities 8. Harmonic numbers and

Introduction to Cluster Algebras

These are notes for a series of lectures presented at the ASIDE conference 2016. The definition of a cluster algebra is motivated through several examples, namely Markov triples, the Grassmannians

Advanced Modern Algebra

This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different