Type sequence
Definition
Let \(S\) be a numerical semigroup, let \(G(S) = \{x_1, \ldots, x_g\}\) be the set of gaps of \(S\) and \(N(S) = \{s_1, \ldots, s_n\}\) the set of left elements of \(S\), both ordered from smallest to larger. For each \(j \in \{1, \ldots, g\}\), it is defined
\[ \sigma_j = \max ~ \{i \in \{1, \ldots, n\} ~ | ~ x_j + s_i \not \in S \}. \]
Then, it is defined the type sequence of \(S\) as the tuple \((t_1(S), \ldots, t_n(S))\), where for all \(i \in \{1, \ldots, g\}\),
\[ t_i(S) = |\{j \in \{1, \ldots, g\} ~ | ~ \sigma_j = i \}|. \]
Since \(s_1 = 0\), it follows that \(\sigma_j\) is well defined. Moreover, from the definition it is deduced that \(t_1(S)\) is the type of \(S\) and its sum is \(\sum_{i = 1}^n t_i(S) = |G(S)|\), the genus of \(S\).
Examples
\(\circ\) For any numerical semigroup with set of gaps \(G(S) = \{x_1, \ldots, x_g\}\) and left elements \(N(S) = \{s_1, \ldots, s_n\}\), both ordered from smallest to larger, it holds that \(t_i(S) \ne 0\) for all \(i \in \{1, \ldots, n\}\). Indeed, for any \(i \in \{1, \ldots, n\}\), it is well known that \(F(S) - s_i \in G(S)\), where \(F(S)\) denotes the Frobenius number of \(S\), and from the definition of Frobenius number, it holds that \(F(S) - s_i + s_{j} \in S\) for all \(j \in \{i+1, \ldots, n\}\). Therefore, taking into account that \(F(S) - s_i + s_i = F(S) \not \in S\), if \(x_j = F(S) - s_i\), it is concluded that \(\sigma_j = i\) and \(t_i(S) \ne 0\).
\(\circ\) Let \(S = \langle 5, 7, 9 \rangle\). Its set of gaps and its set of left elements are \(G(S) = \{1, 2, 3, 4, 6, 8, 11, 13\}\) and \(N(S) = \{0, 5, 7, 9, 10, 12\}\), respectively. With few computations,
\[ \sigma_{13} = 1, ~~ \sigma_{11} = 1, ~~ \sigma_8 = 2, ~~ \sigma_6 = 3, ~~ \sigma_4 = 4, ~~ \sigma_3 = 5, ~~ \sigma_2 = 4, ~~ \sigma_1 = 6, \]
concluding that \((2, 1, 1, 2, 1, 1)\) is the type sequence of \(S\).
Examples with GAP
The following example is made with the package NumericalSgps in GAP.
\(\diamond\) Let \(S = \langle 18, 21, 46, 49, 59, 61, 104 \rangle\), in GAP:
gap> S := NumericalSemigroup(18, 21, 46, 49, 59, 61, 104);
<Numerical semigroup with 7 generators>Given a numerical semigroup \(S\), the functions TypeSequence and TypeSequenceOfNumericalSemigroup return the type sequence of \(S\).
gap> TypeSequence(S);
[ 6, 4, 5, 3, 2, 3, 3, 2, 1, 2, 1, 2, 2, 2, 3, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1,
1, 1, 1 ]
gap> TypeSequenceOfNumericalSemigroup(S) = TypeSequence(S);
trueReferences
https://gap-packages.github.io/ numericalsgps.