The Frobenius problem for Numerical Semigroups generated by binomial coefficients

TL;DR

This work determines the Frobenius number for numerical semigroups generated by binomial coefficients with fixed upper, $S(B_n)$, by deriving explicit Apéry sets from modular-binomial congruences. It provides closed-form Apéry sets and Frobenius numbers for general $n$ (via its prime-factorization) and simplified forms in the prime-power case, along with genus and the pseudo-Frobenius structure; it also proves that $S(B_n)$ is telescopic and complete intersection with type $1$, implying symmetry. The paper connects these number-theoretic results to identities among binomial coefficients and to the theory of $(s,s+1,s+p)$-core partitions, including an algorithm to count admissible core-pairs and concrete data for examples. These contributions advance understanding of binomial-structure-generated semigroups and their combinatorial/partition-theoretic applications. The methods combine Apéry-set techniques, $p$-adic congruences, and explicit generator analysis, yielding concrete, usable formulas and constructive procedures.

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.

Theorems & Definitions (54)

• Theorem 1
• Remark 2
• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Definition 1.6
• Definition 1.7
• Lemma 1.8