The Frobenius problem for Numerical Semigroups generated by binomial coefficients

TL;DR

This paper solves the Frobenius problem for numerical semigroups generated by binomial coefficients, providing new identities and linking results to core partition theory.

Contribution

It introduces a solution to the Frobenius problem specifically for semigroups generated by binomial coefficients, a novel class of problems.

Findings

Derived explicit formulas for the Frobenius number in this context.

Established new identities among binomial coefficients.

Connected the results to the theory of core partitions.

Abstract

The greatest integer that does not belong to a numerical semigroup $S$ is called the Frobenius number of $S$, and finding the Frobenius number is called the Frobenius problem. In this paper, we solve the Frobenius problem for the numerical semigroups generated by binomial coefficients. As applications, we provide some nontrivial identities among binomial coefficients, and we also connect the main results to the theory of $(s,s+1,s+p)$-core partitions of integers.