A Method to construct the Sparse-paving Matroids over a Finite Set

TL;DR

This paper introduces an algorithm for constructing sparse-paving matroids over finite sets, providing bounds on their circuits and a matrix-based method for partitioning subsets to generate such matroids.

Contribution

It presents a novel algorithm for constructing sparse-paving matroids and a matrix-based partitioning method, advancing understanding of their structure and enumeration.

Findings

Derived bounds on the number of circuits in sparse-paving matroids

Proved an asymptotic relation between sparse-paving and all matroids

Introduced a matrix-based partitioning method for subsets

Abstract

In this work we present an algorithm to construct sparse-paving matroids over finite set $S$. From this algorithm we derive some useful bounds on the cardinality of the set of circuits of any Sparse-Paving matroids which allow us to prove in a simple way an asymptotic relation between the class of Sparse-paving matroids and the whole class of matroids. Additionally we introduce a matrix based method which render an explicit partition of the $r$-subsets of $S$, $\binom{S}{r}=\sqcup_{i=1}^{\gamma }\mathcal{U}_{i}$ such that each $\mathcal{U}_{i}$ defines a sparse-paving matroid of rank $r$.