On linear equations arising in Combinatorics (Part I)

TL;DR

This paper introduces a class of integer equations relevant to combinatorics, providing a method to determine solvability through inequality checks, exemplified by Gale-Ryser equations.

Contribution

It presents a new framework for solving certain combinatorial equations by inequalities, extending understanding of Gale-Ryser type problems.

Findings

Equations can be checked for solutions via inequalities.

The approach simplifies solving combinatorial equations.

Provides a unified perspective on Gale-Ryser equations.

Abstract

The main point of this paper is to present a class of equations over integers that one can check if they have a solution by checking a set of inequalities. The prototype of such equations is the equations appearing in the well-known Gale-Ryser theorem.