Let \(\Lambda\) be a numerical semigroup, that is, a cofinite submonoid of \((\mathbb{N},+)\). Write \(\Lambda=\{\lambda_0,\lambda_1,\dots\}\) with \(\lambda_i< \lambda_{i+1}\) for all \(i\in \mathbb{N}\) and \(\lambda_0=0\). The set \(\mathbf{N}\setminus \Lambda\) is the set of gaps of \(\Lambda\), and its cardinality is known as the genus of \(\Lambda\), denoted \(\operatorname{g}(\Lambda)\). The least integer \(c\) such that \(c+\mathbb{N}\subseteq \Lambda\) is called the conductor of \(\Lambda\), which is denoted by \(\operatorname{c}(\Lambda)\). The rank of \(\Lambda\) is defined as \(\operatorname{k}(\Lambda)=\operatorname{c}(\Lambda)-\operatorname{g}(\Lambda)\). The Frobenius number of \(\Lambda\) is \(\operatorname{F}(\Lambda)=\max(\mathbf{Z}\setminus \Lambda)\); it turns out that \(\operatorname{c}(\Lambda)=\operatorname{F}(\Lambda)+1\). The elements in \(\Lambda\) that are smaller than the conductor of \(\Lambda\) are known left elements of \(\Lambda\). The set of left elements of \(\Lambda\) is denoted by \(\operatorname{L}(\Lambda)\).
The set of gaps of \(\Lambda\) can be encoded as a the bitstream \(\operatorname{G}(\Lambda)=g_0g_1\dots g_{c-1}\), with \(c=\operatorname{c}(\Lambda)\) and \(g_i=0\) if \(i+1\in \Lambda\) and \(g_i=0\) otherwise.
For \(p< \operatorname{k}(\Lambda)\), an element \(\lambda_s\ge \operatorname{c}(\Lambda)\) is said to be an order-\(p\) seed of \(\Lambda\) if \(\lambda_s+\lambda_p\neq \lambda_i+\lambda_j\) for any integers \(i\) and \(j\) with \(p< i \le j < s\). One can encode the seeds of \(\Lambda\) as a bitstream \(\operatorname{S}(\Lambda)=S_0S_1\dots S_{c-1}\) where \(S_i=1\) if \(c+i-\lambda_j\) is an order-\(j\) seed of \(\Lambda\), where \(j\) is the unique integer such that \(\lambda_j\le i< \lambda_{j+1}\), and \(S_i=0\) otherwise. Seeds have been a central tool to compute the set of all numerical semigroups up to a certain genus. Minimal generators of \(\Lambda\) (elements that are not sum of two non-zero elements) larger than or equal to the conductor \(c\) are called right generators. These correspond to order-0 seeds. If \(\lambda_s\) is a right generator, then \(\Lambda\setminus\{\lambda_s\}\) is a numerical semigroup with genus \(\operatorname{g}(\Lambda)+1\) and \(\operatorname{F}(\Lambda\setminus\{\lambda_s\})=\lambda_s\). In this setting, we say that \(\Lambda\setminus\{\lambda_s\}\) is a child of \(\Lambda\). In this way, one can construct the tree of all numerical semigroups.
The author presents a new bitstream which encodes both the gap bitstream and the seeds bitstream (it is almost a concatenation of both). This bitstream is denoted by \(\Sigma(\Lambda)=\Sigma_0\Sigma_1\dots \Sigma_{2c-1}\), where \(\Sigma_i=0\) if \(i\in \operatorname{L}(\Lambda)+\operatorname{L}(\Lambda)\), and \(\Sigma_i=1\) otherwise.
With this new encoding, the author describes the gap and seeds bitstreams of numerical semigroups of rank up to three, and shows how the bitstream \(\Sigma\) is updated from parent to children.
Set \(m=\lambda_1\) (the multiplicity of \(\Lambda\)), \(u=\lambda_2-\lambda_1\) and \(v=\lambda_3-\lambda_2\). The author shows how to recover the sets of all children, grandchildren and great-grandchildren of \(\Lambda\) in the tree of numerical semigroups from \(\Sigma(\Lambda)\), \(m\), \(u\) and \(v\).
With all this, the author presents a new version of the seeds algorithm that is faster than the previous known algorithms to compute the tree of numerical semigroups up to a certain genus. The new algorithm is then used to find new examples of (non-)Eliahou semigroups, and to check that Wilf’s conjecture holds up to genus 66.
The author presents many illustrating examples, figures, tables showing execution times, and the limitations of the algorithm when using 128 bit integers.