Abstract
We provide a new way to represent numerical semigroups by showing that the position of every Apéry set of a numerical semigroup \(S\) in the enumeration of the elements of \(S\) is unique, and that \(S\) can be re-constructed from this “position vector.” We extend the discussion to more general objects called numerical sets, and show that there is a one-to-one correspondence between \(m\)-tuples of positive integers and the position vectors of numerical sets closed under addition by \(m+1\). We consider the problem of determining which position vectors correspond to numerical semigroups.
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