• Corpus ID: 124485523

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. 

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=

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