GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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<?xml version="1.0" encoding="UTF-8"?>
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<Section>
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  <Heading>Wilf's conjecture</Heading>
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  Let <M>S</M> be a numerical semigroup, with conductor <M>c</M> and embedding dimension <M>e</M>. Denote by <M>l</M> the cardinality of the set of elements in  <M>S</M> smaller than <M>c</M>. Wilf in <Cite Key="Wilf"></Cite> asked whether or not <M>l/c\ge 1/e</M> for all numerical semigroups. In this section we give some functions to experiment with this conjecture, as defined in <Cite Key="E"></Cite>.
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  <ManSection>
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    <Attr Name="WilfNumber" Arg="S" Label="for numerical semigroup"/>
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    <Attr Name="WilfNumberOfNumericalSemigroup" Arg="S"/>
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    <Description>
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      <C>S</C> 
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      is a numerical semigroup.  Let <M>c</M>, <M>e</M> and <M>l</M> be the conductor, embedding dimension and number of  elements smaller than <M>c</M> in <A>S</A>. Returns <M>e l-c</M>, which was conjetured by Wilf to be nonnegative.
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      <Example><![CDATA[
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gap> l:=NumericalSemigroupsWithGenus(10);;
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gap> Filtered(l, s->WilfNumberOfNumericalSemigroup(s)<0); 
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[  ]
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gap> Maximum(Set(l, s->WilfNumberOfNumericalSemigroup(s)));
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70
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gap> s := NumericalSemigroup(13,25,37);;
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gap> WilfNumber(s);                     
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96
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]]></Example>
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    </Description>
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  </ManSection>
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  <ManSection>
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    <Attr Name="EliahouNumber" Arg="S" Label="for numerical semigroup"/>
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    <Attr Name="TruncatedWilfNumberOfNumericalSemigroup" Arg="S"/>
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    <Description>
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      <C>S</C> 
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      is a numerical semigroup.  Let <M>c</M>, <M>m</M>, <M>s</M> and <M>l</M> be the conductor, multiplicity, number of generators smaller than <M>c</M>, and number of  elements smaller than <M>c</M> in <A>S</A>, respectively. Let <M>q</M> and <M>r</M> be the quotient and negative remainder of the division of <M>c</M> by <M>m</M>, that is, <M>c=qm-r</M>. Returns <M>s l-qd_q+r</M>, where <M>d_q</M> corresponds with the number of integers in <M>[c,c+m[</M> that are not minimal generators of <A>S</A>.
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      <Example><![CDATA[
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gap> s:=NumericalSemigroup(5,7,9);;
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gap> TruncatedWilfNumberOfNumericalSemigroup(s);
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gap> s:=NumericalSemigroupWithGivenElementsAndFrobenius([14,22,23],55);;
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gap> EliahouNumber(s);
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-1
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]]></Example>
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    </Description>
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  </ManSection>
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  <ManSection>
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    <Attr Name="ProfileOfNumericalSemigroup" Arg="S"/>
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    <Description>
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      <C>S</C> 
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      is a numerical semigroup.  Let <M>c</M> and <M>m</M> be the conductor and  multiplicity of <A>S</A>, respectively. Let <M>q</M> and <M>r</M> be the quotient and nonpositive remainder of the division of <M>c</M> by <M>m</M>, that is, <M>c=qm-r</M>.  Returns a list of lists of integers, each list is the cardinality of <M> S \cap [jm-r, (j+1)m-r[</M> with <M> j</M> in [1..q-1].
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      <Example><![CDATA[
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gap> s:=NumericalSemigroup(5,7,9);;
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gap> ProfileOfNumericalSemigroup(s);
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[ 2, 1 ]
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gap> s:=NumericalSemigroupWithGivenElementsAndFrobenius([14,22,23],55);;
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gap> ProfileOfNumericalSemigroup(s);
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[ 3, 0, 0 ]
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]]></Example>
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    </Description>
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  </ManSection>
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  <ManSection>
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    <Attr Name="EliahouSlicesOfNumericalSemigroup" Arg="S"/>
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    <Description>
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      <C>S</C> 
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      is a numerical semigroup.  Let <M>c</M> and <M>m</M> be the conductor and  multiplicity of <A>S</A>, respectively. Let <M>q</M> and <M>r</M> be the quotient and negative remainder of the division of <M>c</M> by <M>m</M>, that is, <M>c=qm-r</M>.  Returns a list of lists of integers, each list is the set <M> S \cap [jm-r, (j+1)m-r[</M> with <M> j</M> in [1..q]. So this is a partition of the set of small elements of <A>S</A> (without <M>0</M>).
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      <Example><![CDATA[
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gap> s:=NumericalSemigroup(5,7,9);;                                     
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gap> EliahouSlicesOfNumericalSemigroup(s);
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[ [ 5, 7 ], [ 9, 10, 12 ] ]
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gap> SmallElements(s);
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[ 0, 5, 7, 9, 10, 12, 14 ]
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]]></Example>
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    </Description>
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  </ManSection>
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</Section>
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