Wilf's conjecture Let S be a numerical semigroup, with conductor c and embedding dimension e. Denote by l the cardinality of the set of elements in S smaller than c. Wilf in asked whether or not l/c\ge 1/e for all numerical semigroups. In this section we give some functions to experiment with this conjecture, as defined in . S is a numerical semigroup. Let c, e and l be the conductor, embedding dimension and number of elements smaller than c in S. Returns e l-c, which was conjetured by Wilf to be nonnegative.
gap> l:=NumericalSemigroupsWithGenus(10);; gap> Filtered(l, s->WilfNumberOfNumericalSemigroup(s)<0); [ ] gap> Maximum(Set(l, s->WilfNumberOfNumericalSemigroup(s))); 70 gap> s := NumericalSemigroup(13,25,37);; gap> WilfNumber(s); 96
S is a numerical semigroup. Let c, m, s and l be the conductor, multiplicity, number of generators smaller than c, and number of elements smaller than c in S, respectively. Let q and r be the quotient and negative remainder of the division of c by m, that is, c=qm-r. Returns s l-qd_q+r, where d_q corresponds with the number of integers in [c,c+m[ that are not minimal generators of S.
gap> s:=NumericalSemigroup(5,7,9);; gap> TruncatedWilfNumberOfNumericalSemigroup(s); 4 gap> s:=NumericalSemigroupWithGivenElementsAndFrobenius([14,22,23],55);; gap> EliahouNumber(s); -1
S is a numerical semigroup. Let c and m be the conductor and multiplicity of S, respectively. Let q and r be the quotient and nonpositive remainder of the division of c by m, that is, c=qm-r. Returns a list of lists of integers, each list is the cardinality of S \cap [jm-r, (j+1)m-r[ with j in [1..q-1].
gap> s:=NumericalSemigroup(5,7,9);; gap> ProfileOfNumericalSemigroup(s); [ 2, 1 ] gap> s:=NumericalSemigroupWithGivenElementsAndFrobenius([14,22,23],55);; gap> ProfileOfNumericalSemigroup(s); [ 3, 0, 0 ]
S is a numerical semigroup. Let c and m be the conductor and multiplicity of S, respectively. Let q and r be the quotient and negative remainder of the division of c by m, that is, c=qm-r. Returns a list of lists of integers, each list is the set S \cap [jm-r, (j+1)m-r[ with j in [1..q]. So this is a partition of the set of small elements of S (without 0).
gap> s:=NumericalSemigroup(5,7,9);; gap> EliahouSlicesOfNumericalSemigroup(s); [ [ 5, 7 ], [ 9, 10, 12 ] ] gap> SmallElements(s); [ 0, 5, 7, 9, 10, 12, 14 ]