Related papers: An Introduction to Combinatorics of Determinacy
The twenty-first century is a data-driven era where human activities and behavior, physical phenomena, scientific discoveries, technology advancements, and almost everything that happens in the world resulting in massive generation,…
We provide a new foundation for combinatorial commutative algebra and Stanley-Reisner theory using the partition complex introduced in [Adi18]. One of the main advantages is that it is entirely self-contained, using only a minimal knowledge…
In this article, we discuss the identity and indistinguishability of quantum systems and the consequent need to introduce an extra postulate in Quantum Mechanics to correctly describe situations involving indistinguishable particles. This…
I present a bijection on integer partitions that leads to recursive expressions, closed formulae and generating functions for the cardinality of certain sets of partitions of a positive integer $n$. The bijection leads also to a product on…
An archetypal problem discussed in computer science is the problem of searching for a given number in a given set of numbers. Other than sequential search, the classic solution is to sort the list of numbers and then apply binary search.…
In this paper we find exact formulas for the numbers of partitions and compositions of an element into $m$ parts over a finite field, i.e. we find the number of nonzero solutions of the equation $x_1+x_2+...+x_m=z$ over a finite field when…
Integer partitions are one of the most fundamental objects of combinatorics (and number theory), and so is enumerating objects avoiding patterns. In the present paper we describe two approaches for the systematic counting of classes of…
Evaluating joint probabilities of potential outcomes and observed variables, and their linear combinations, is a fundamental challenge in causal inference. This paper addresses the bounding and identification of these probabilities in…
In this article we will derive a combinatorial formula for the partition function p(n). In the second part of the paper we will establish connection between partitions and q-binomial coefficients and give new interpretation for q-binomial…
A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and number-theoretical nature.
Recently there has been several works estimating the number of $n\times n$ matrices with elements from some finite sets $\mathcal X$ of arithmetic interest and of a given determinant. Typically such results are compared with the trivial…
This paper proposes new derivations of three well-known sorting algorithms, in their functional formulation. The approach we use is based on three main ingredients: first, the algorithms are derived from a simpler algorithm, i.e. the…
We translate Uchimura's identity for the divisor function and whose generalizations into combinatorics of partitions, and give a combinatorial proof of them. As a by-product of their proofs, we obtain some combinatorial results.
This paper considers the properties of Tribonacci numbers on identities, matrices, and determinants. In the first front part, we obtain several symmetric identities of Tribonacci numbers by a matrix-based approach and binomial inversion…
We present a new partition identity and give a combinatorial proof of our result. This generalizes a result of Andrew's in which he considers the generation function for partitions with respect to size, number of odd parts, and number of…
This article is based on a series of lectures on toric varieties given at RIMS, Kyoto. We start by introducing toric varieties, their basic properties and later pass to more advanced topics relating mostly to combinatorics.
Although symmetry methods and analysis are a necessary ingredient in every physicist's toolkit, rather less use has been made of combinatorial methods. One exception is in the realm of Statistical Physics, where the calculation of the…
We give an introduction to some of the recent ideas that go under the name "geometric complexity theory". We first sketch the proof of the known upper and lower bounds for the determinantal complexity of the permanent. We then introduce the…
We give short elementary expositions of combinatorial proofs of some variants of Euler's partitition problem that were first addressed analytically by George Andrews, and later combinatorially by others. Our methods, based on ideas from a…
In this article, we present a short, non-exhaustive study of an important and well-known property of combinatorial sequences - unimodality. We shall have a look at a sample of classical results on unimodality and related properties, and…