ExampleMagmaCode9.mgm

// Want to find the "distinct" necklaces
// with n black/white beads.
// Two necklaces are the "same"
// if an element of D8 changes one to the other.
// Represent a necklace by the set of black beads.

n := 8;

Dn := DihedralGroup(n);
DistinctNecklaces := {};

k := 3;

Nk := Subsets({1..n}, k);
while not IsEmpty(Nk) do
  t := Rep(Nk);
  Include(~DistinctNecklaces, t);
  O := Orbit(Dn,t);
  for o in O do
    Exclude(~Nk,o);
  end for;
end while;

DistinctNecklaces;


function Necklaces(n,k)
  // Return necklaces with n total beads, k black
  Dn := DihedralGroup(n);
  DistinctNecklaces := {};

  Nk := Subsets({1..n}, k);
  while not IsEmpty(Nk) do
    t := Rep(Nk);
    Include(~DistinctNecklaces, t);
    O := Orbit(Dn,t);
    for o in O do
      Exclude(~Nk,o);
    end for;
  end while;

  return DistinctNecklaces;
end function;


function Necklaces(n)
  // Return all necklaces with n total
  // black/white beads
  Dn := DihedralGroup(n);
  DistinctNecklaces := {};
  S := {1..n};
  for k in [0..n] do
    Nk := Subsets(S, k);
    while not IsEmpty(Nk) do
      t := Rep(Nk);
      Include(~DistinctNecklaces, t);
      O := Orbit(Dn,t);
      for o in O do
        Exclude(~Nk,o);
      end for;
    end while;
  end for;

  return DistinctNecklaces;
end function;


// For 3 colors: (copied from Necklaces(n,k))
function Necklaces(n, k, r)
  // return necklaces with n beads of 3 colors
  // k color '1', r of color '2', n-(k+r) color '0'

  // n := 8; k := 2; r := 1;
  Dn := DihedralGroup(n);
  DistinctNecklaces := {};

  //Need to define the set Nkr
  S := {1..n};
  Nkr := {};
  Sk := Subsets(S,k);
  for s in Sk do
    Sr := Subsets(S diff s, r);
    for t in Sr do
      Include(~Nkr, [s,t]);
    end for;
  end for;


  while not IsEmpty(Nkr) do
    t := Rep(Nkr);
    Include(~DistinctNecklaces, t);
    O := Orbit(Dn,t);
    for o in O do
      Exclude(~Nkr,o);
    end for;
  end while;

  return DistinctNecklaces;
end function;

// Possible ways to loop through values of k,r.
for k in [0..n] do
  for r in [0..n-k] do

  end for;
end for;

for k,r in [0..n] do
  if k+r gt n then
    continue;
  end if;

end for;





G := Graph< 5 | {1,2}, {2,3}, {3,4}, {4,5}, {5,1} >;
G;
Vertices(G);
Edges(G);
V := VertexSet(G);
V;
1;
V!1;
1 eq V!1;
Type(1);
Type(V!1);
Sub3 := Subsets({1,2,3,4,5}, 3);
r := Random(Sub3);
r;
AssignVertexLabels(~G, [1,1,1,0,0]);
G;
Label(V!1);
IsVertexLabelled(V!1);
IsLabelled(V!1);
V := VertexSet(G);
IsLabelled(V!1);
Label(VertexSet(G)!1);
G := Graph< 5 | {1,2}, {2,3}, {3,4}, {4,5}, {5,1} >;
V := VertexSet(G);
AssignVertexLabels(~G, [1,1,1,0,0]);
V := VertexSet(G);
n := 10;
k := 4;
r := 2;
S1 := Subsets({1..10}, k);
s1 := Random(S1);
s1;
S2 := Subsets( {1..10} - s1 , r);
S2 := Subsets( {1..10} diff s1 , r);
{1..10} diff s1;
s2 := Random(S2);
s1;
s2;
A := AutomorphismGroup(G);
G := Graph< 5 | {1,2}, {2,3}, {3,4}, {4,5}, {5,1} >;
1;2Ak := 4;
V := VertexSet(G);
A := AutomorphismGroup(G);
s := [{V!1, V!2}, {V!3}];
Orbit(A,s);
Orbit(A, V!1);
A;
s := [{1, 2}, {3}];
Orbit(A,s);
Nkr := {};
Sk := Subsets({1..8}, 3);
for sk in Sk do
Sr := Subsets({1..8} diff sk, 2);
for sr in Sr do
Include(~Nkr, [sk,sr]);
end for;
end for;
#Nkr;
Nkr;
{ N : N in Nkr | N[1] eq {1,2,3} };
t := Random(Nkr);
t;
D8 := DihedralGroup(8);
Orbit(D8, t);
DistinctNecklaces := {};
#Nkr;
while not IsEmpty(Nkr) do
  t := Rep(Nkr);
  Include(~DistinctNecklaces, t);
  O := Orbit(D8,t);
  for o in O do
    Exclude(~Nkr,o);
  end for;
end while;
Nkr;
DistinctNecklaces;
#DistinctNecklaces;
Subsets(3,2);
function Necklaces(n,k)
  // Return necklaces with n total beads, k black
  Dn := DihedralGroup(n);
  DistinctNecklaces := {};
  Nk := Subsets({1..n}, k);
  while not IsEmpty(Nk) do
    t := Rep(Nk);
    Include(~DistinctNecklaces, t);
    O := Orbit(Dn,t);
    for o in O do
      Exclude(~Nk,o);
    end for;
  end while;
  return DistinctNecklaces;
end function;
Necklaces(8,3);
n := 8; k := 2; r := 1;
  Dn := DihedralGroup(n);
  DistinctNecklaces := {};
  //Need to define the set Nkr
  S := {1..n};
  Nkr := {};
  Sk := Subsets(S,k)
  for s in Sk do
    Sr := Subsets(S diff s, r);
    for t in Sr do
      Include(~Nkr, [s,t]);
    end for;
  end for;
n := 8; k := 2; r := 1;
  Dn := DihedralGroup(n);
  DistinctNecklaces := {};
S := {1..n};
  Nkr := {};
  Sk := Subsets(S,k);
for s in Sk do
    Sr := Subsets(S diff s, r);
    for t in Sr do
      Include(~Nkr, [s,t]);
    end for;
  end for;
Nkr;
while not IsEmpty(Nkr) do
    t := Rep(Nkr);
    Include(~DistinctNecklaces, t);
    O := Orbit(Dn,t);
    for o in O do
      Exclude(~Nkr,o);
    end for;
  end while;
DistinctNecklaces;
function Necklaces(n, k, r)
  // return necklaces with n beads of 3 colors
  // k color '1', r of color '2', n-(k+r) color '0'
  // n := 8; k := 2; r := 1;
  Dn := DihedralGroup(n);
  DistinctNecklaces := {};
  //Need to define the set Nkr
  S := {1..n};
  Nkr := {};
  Sk := Subsets(S,k);
  for s in Sk do
    Sr := Subsets(S diff s, r);
    for t in Sr do
      Include(~Nkr, [s,t]);
    end for;
  end for;
  while not IsEmpty(Nkr) do
    t := Rep(Nkr);
    Include(~DistinctNecklaces, t);
    O := Orbit(Dn,t);
    for o in O do
      Exclude(~Nkr,o);
    end for;
  end while;
  return DistinctNecklaces;
end function;
Necklaces(6, 2,2);