# Notes on the combinatorial fundamentals of algebra

@inproceedings{Grinberg2016NotesOT, title={Notes on the combinatorial fundamentals of algebra}, author={Darij Grinberg}, year={2016} }

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.

## 28 Citations

### The 4-periodic spiral determinant

- Mathematics
- 2019

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

- Mathematics
- 2021

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

- Mathematics
- 2022

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

- Mathematics2021 60th IEEE Conference on Decision and Control (CDC)
- 2021

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

- MathematicsFunctional Analysis and Its Applications
- 2021

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

- Mathematics
- 2022

. 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

- Mathematics
- 2020

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

- Mathematics
- 2016

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

- Mathematics
- 2019

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

- Mathematics
- 2017

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=…

## References

SHOWING 1-10 OF 154 REFERENCES

### An Introduction to Abstract Algebra

- Mathematics, Computer Science
- 2005

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

- Mathematics
- 2007

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

- Mathematics
- 2001

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

- MathematicsAdv. Appl. Math.
- 2002

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.

### SOME RESULTS FOR SUMS OF THE INVERSES OF BINOMIAL COEFFICIENTS

- Mathematics
- 2005

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

- Mathematics
- 2003

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

### Hopf Algebras in Combinatorics

- Mathematics
- 2014

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

- Mathematics
- 2003

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

- Mathematics
- 2017

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

- Mathematics
- 2002

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…