---input---
/*++ $Id: AlternatingGroup.mu,v 1.4 2003/09/08 15:00:47 nthiery Exp $

Dom::AlternatingGroup(n) -- the Alternating Group of {1..n}

n	   - integer >= 1

Elements are represented as in Dom::PermutationGroup(n)

Author:	     Nicolas M. Thiéry <nthiery@users.sourceforge.net>
License:     LGPL
Created:     August 8th, 1999
Last update: $Date: 2003/09/08 15:00:47 $
++*/

domain Dom::AlternatingGroup(n: Type::PosInt)
    inherits Dom::PermutationGroup(n,toBeDefined);
    category Cat::PermutationGroup;
    axiom Ax::canonicalRep;

/*--
    size

    Size of the group.
--*/

    size := fact(n)/2;

/*--
    generators

    A list of generators of the group

    The first 3-cycle (1,2,3), and a maximal even cycle (1,...,n) or
    (2,...,n) depending on the parity of n

--*/

    generators :=
    if	 n<=2	     then generators:=[dom([[1]])];
    elif n=3	     then generators:=[dom([[1,2,3]])];
    elif n mod 2=0   then generators:=[dom([[1,2,3]]), dom([[$2..n]])];
    else		  generators:=[dom([[1,2,3]]), dom([[$1..n]])];
    end_if;
    
/*--
    allElements

    List of all the elements of the group
--*/

    allElements :=
    proc()
	option remember;
	local p;
    begin
	[new(dom,p) $ p in select(combinat::permutations(n),
				  p->bool(combinat::permutations::sign(p)=1))];
    end_proc;

/*--
    cycleTypes:

    Count the elements of the group by cycle type.
    (Cf Cat::PermutationGroupModule).

    Same algorithm as for Dom::SymmetricGroup, but only even permutations
    are considered. This is done by disregarding partitions p such
    that n-length(p) is odd.
--*/

    cycleTypes :=
    proc()
	option remember;
	local t, p, gen;
    begin
	userinfo(3, "cycleTypes: starting computation");
	t:=table();

	gen := combinat::partitions::generator(n);
	while (p:=gen()) <> FAIL do
	    userinfo(5, "working on partition", p);
	    if(n-nops(p) mod 2=0) then
		// Compute the size of the conjugacy class of Sn indexed by p
		// and the cycle type of a permutation in this conjugacy class
                t[combinat::partitions::toExp(p,n)]
                  := combinat::partitions::conjugacyClassSize(p);
	    end_if;
        end_while;
	t;
    end_proc;

begin
    if testargs() then
	if args(0) <> 1 then error("wrong no of args"); end_if;
	if not testtype(n,DOM_INT) then
	    error("argument must be integer")
	end_if;
	if n < 1 then
	    error("argument must be positive")
	end_if;
    end_if;
end_domain:

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')'           Punctuation
']'           Punctuation
';'           Operator
'\n'          Text

' '           Text
' '           Text
' '           Text
' '           Text
'end_if'      Keyword
';'           Operator
'\n'          Text

' '           Text
' '           Text
' '           Text
' '           Text
'\n'          Text

'/*'          Comment.Multiline
'-'           Comment.Multiline
'-'           Comment.Multiline
'\n'          Comment.Multiline

' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
'a'           Comment.Multiline
'l'           Comment.Multiline
'l'           Comment.Multiline
'E'           Comment.Multiline
'l'           Comment.Multiline
'e'           Comment.Multiline
'm'           Comment.Multiline
'e'           Comment.Multiline
'n'           Comment.Multiline
't'           Comment.Multiline
's'           Comment.Multiline
'\n'          Comment.Multiline

'\n'          Comment.Multiline

' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
'L'           Comment.Multiline
'i'           Comment.Multiline
's'           Comment.Multiline
't'           Comment.Multiline
' '           Comment.Multiline
'o'           Comment.Multiline
'f'           Comment.Multiline
' '           Comment.Multiline
'a'           Comment.Multiline
'l'           Comment.Multiline
'l'           Comment.Multiline
' '           Comment.Multiline
't'           Comment.Multiline
'h'           Comment.Multiline
'e'           Comment.Multiline
' '           Comment.Multiline
'e'           Comment.Multiline
'l'           Comment.Multiline
'e'           Comment.Multiline
'm'           Comment.Multiline
'e'           Comment.Multiline
'n'           Comment.Multiline
't'           Comment.Multiline
's'           Comment.Multiline
' '           Comment.Multiline
'o'           Comment.Multiline
'f'           Comment.Multiline
' '           Comment.Multiline
't'           Comment.Multiline
'h'           Comment.Multiline
'e'           Comment.Multiline
' '           Comment.Multiline
'g'           Comment.Multiline
'r'           Comment.Multiline
'o'           Comment.Multiline
'u'           Comment.Multiline
'p'           Comment.Multiline
'\n'          Comment.Multiline

'-'           Comment.Multiline
'-'           Comment.Multiline
'*/'          Comment.Multiline
'\n'          Text

'\n'          Text

' '           Text
' '           Text
' '           Text
' '           Text
'allElements' Name.Variable
' '           Text
':'           Operator
'='           Operator
'\n'          Text

' '           Text
' '           Text
' '           Text
' '           Text
'proc'        Keyword
'('           Punctuation
')'           Punctuation
'\n'          Text

'\t'          Text
'option'      Keyword
' '           Text
'remember'    Name.Variable
';'           Operator
'\n'          Text

'\t'          Text
'local'       Keyword
' '           Text
'p'           Name.Variable
';'           Operator
'\n'          Text

' '           Text
' '           Text
' '           Text
' '           Text
'begin'       Keyword
'\n'          Text

'\t'          Text
'['           Punctuation
'new'         Name.Function
'('           Punctuation
'dom'         Name.Builtin.Pseudo
','           Operator
'p'           Name.Variable
')'           Punctuation
' '           Text
'$'           Operator
' '           Text
'p'           Name.Variable
' '           Text
'in'          Operator.Word
' '           Text
'select'      Name.Function
'('           Punctuation
'combinat::permutations' Name.Function
'('           Punctuation
'n'           Name.Variable
')'           Punctuation
','           Operator
'\n'          Text

'\t'          Text
'\t'          Text
'\t'          Text
'\t'          Text
' '           Text
' '           Text
'p'           Name.Variable
'-'           Operator
'>'           Operator
'bool'        Name.Function
'('           Punctuation
'combinat::permutations::sign' Name.Function
'('           Punctuation
'p'           Name.Variable
')'           Punctuation
'='           Operator
'1'           Literal.Number
')'           Punctuation
')'           Punctuation
']'           Punctuation
';'           Operator
'\n'          Text

' '           Text
' '           Text
' '           Text
' '           Text
'end_proc'    Keyword
';'           Operator
'\n'          Text

'\n'          Text

'/*'          Comment.Multiline
'-'           Comment.Multiline
'-'           Comment.Multiline
'\n'          Comment.Multiline

' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
'c'           Comment.Multiline
'y'           Comment.Multiline
'c'           Comment.Multiline
'l'           Comment.Multiline
'e'           Comment.Multiline
'T'           Comment.Multiline
'y'           Comment.Multiline
'p'           Comment.Multiline
'e'           Comment.Multiline
's'           Comment.Multiline
':'           Comment.Multiline
'\n'          Comment.Multiline

'\n'          Comment.Multiline

' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
'C'           Comment.Multiline
'o'           Comment.Multiline
'u'           Comment.Multiline
'n'           Comment.Multiline
't'           Comment.Multiline
' '           Comment.Multiline
't'           Comment.Multiline
'h'           Comment.Multiline
'e'           Comment.Multiline
' '           Comment.Multiline
'e'           Comment.Multiline
'l'           Comment.Multiline
'e'           Comment.Multiline
'm'           Comment.Multiline
'e'           Comment.Multiline
'n'           Comment.Multiline
't'           Comment.Multiline
's'           Comment.Multiline
' '           Comment.Multiline
'o'           Comment.Multiline
'f'           Comment.Multiline
' '           Comment.Multiline
't'           Comment.Multiline
'h'           Comment.Multiline
'e'           Comment.Multiline
' '           Comment.Multiline
'g'           Comment.Multiline
'r'           Comment.Multiline
'o'           Comment.Multiline
'u'           Comment.Multiline
'p'           Comment.Multiline
' '           Comment.Multiline
'b'           Comment.Multiline
'y'           Comment.Multiline
' '           Comment.Multiline
'c'           Comment.Multiline
'y'           Comment.Multiline
'c'           Comment.Multiline
'l'           Comment.Multiline
'e'           Comment.Multiline
' '           Comment.Multiline
't'           Comment.Multiline
'y'           Comment.Multiline
'p'           Comment.Multiline
'e'           Comment.Multiline
'.'           Comment.Multiline
'\n'          Comment.Multiline

' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
'('           Comment.Multiline
'C'           Comment.Multiline
'f'           Comment.Multiline
' '           Comment.Multiline
'C'           Comment.Multiline
'a'           Comment.Multiline
't'           Comment.Multiline
':'           Comment.Multiline
':'           Comment.Multiline
'P'           Comment.Multiline
'e'           Comment.Multiline
'r'           Comment.Multiline
'm'           Comment.Multiline
'u'           Comment.Multiline
't'           Comment.Multiline
'a'           Comment.Multiline
't'           Comment.Multiline
'i'           Comment.Multiline
'o'           Comment.Multiline
'n'           Comment.Multiline
'G'           Comment.Multiline
'r'           Comment.Multiline
'o'           Comment.Multiline
'u'           Comment.Multiline
'p'           Comment.Multiline
'M'           Comment.Multiline
'o'           Comment.Multiline
'd'           Comment.Multiline
'u'           Comment.Multiline
'l'           Comment.Multiline
'e'           Comment.Multiline
')'           Comment.Multiline
'.'           Comment.Multiline
'\n'          Comment.Multiline

'\n'          Comment.Multiline

' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
'S'           Comment.Multiline
'a'           Comment.Multiline
'm'           Comment.Multiline
'e'           Comment.Multiline
' '           Comment.Multiline
'a'           Comment.Multiline
'l'           Comment.Multiline
'g'           Comment.Multiline
'o'           Comment.Multiline
'r'           Comment.Multiline
'i'           Comment.Multiline
't'           Comment.Multiline
'h'           Comment.Multiline
'm'           Comment.Multiline
' '           Comment.Multiline
'a'           Comment.Multiline
's'           Comment.Multiline
' '           Comment.Multiline
'f'           Comment.Multiline
'o'           Comment.Multiline
'r'           Comment.Multiline
' '           Comment.Multiline
'D'           Comment.Multiline
'o'           Comment.Multiline
'm'           Comment.Multiline
':'           Comment.Multiline
':'           Comment.Multiline
'S'           Comment.Multiline
'y'           Comment.Multiline
'm'           Comment.Multiline
'm'           Comment.Multiline
'e'           Comment.Multiline
't'           Comment.Multiline
'r'           Comment.Multiline
'i'           Comment.Multiline
'c'           Comment.Multiline
'G'           Comment.Multiline
'r'           Comment.Multiline
'o'           Comment.Multiline
'u'           Comment.Multiline
'p'           Comment.Multiline
','           Comment.Multiline
' '           Comment.Multiline
'b'           Comment.Multiline
'u'           Comment.Multiline
't'           Comment.Multiline
' '           Comment.Multiline
'o'           Comment.Multiline
'n'           Comment.Multiline
'l'           Comment.Multiline
'y'           Comment.Multiline
' '           Comment.Multiline
'e'           Comment.Multiline
'v'           Comment.Multiline
'e'           Comment.Multiline
'n'           Comment.Multiline
' '           Comment.Multiline
'p'           Comment.Multiline
'e'           Comment.Multiline
'r'           Comment.Multiline
'm'           Comment.Multiline
'u'           Comment.Multiline
't'           Comment.Multiline
'a'           Comment.Multiline
't'           Comment.Multiline
'i'           Comment.Multiline
'o'           Comment.Multiline
'n'           Comment.Multiline
's'           Comment.Multiline
'\n'          Comment.Multiline

' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
'a'           Comment.Multiline
'r'           Comment.Multiline
'e'           Comment.Multiline
' '           Comment.Multiline
'c'           Comment.Multiline
'o'           Comment.Multiline
'n'           Comment.Multiline
's'           Comment.Multiline
'i'           Comment.Multiline
'd'           Comment.Multiline
'e'           Comment.Multiline
'r'           Comment.Multiline
'e'           Comment.Multiline
'd'           Comment.Multiline
'.'           Comment.Multiline
' '           Comment.Multiline
'T'           Comment.Multiline
'h'           Comment.Multiline
'i'           Comment.Multiline
's'           Comment.Multiline
' '           Comment.Multiline
'i'           Comment.Multiline
's'           Comment.Multiline
' '           Comment.Multiline
'd'           Comment.Multiline
'o'           Comment.Multiline
'n'           Comment.Multiline
'e'           Comment.Multiline
' '           Comment.Multiline
'b'           Comment.Multiline
'y'           Comment.Multiline
' '           Comment.Multiline
'd'           Comment.Multiline
'i'           Comment.Multiline
's'           Comment.Multiline
'r'           Comment.Multiline
'e'           Comment.Multiline
'g'           Comment.Multiline
'a'           Comment.Multiline
'r'           Comment.Multiline
'd'           Comment.Multiline
'i'           Comment.Multiline
'n'           Comment.Multiline
'g'           Comment.Multiline
' '           Comment.Multiline
'p'           Comment.Multiline
'a'           Comment.Multiline
'r'           Comment.Multiline
't'           Comment.Multiline
'i'           Comment.Multiline
't'           Comment.Multiline
'i'           Comment.Multiline
'o'           Comment.Multiline
'n'           Comment.Multiline
's'           Comment.Multiline
' '           Comment.Multiline
'p'           Comment.Multiline
' '           Comment.Multiline
's'           Comment.Multiline
'u'           Comment.Multiline
'c'           Comment.Multiline
'h'           Comment.Multiline
'\n'          Comment.Multiline

' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
' '           Comment.Multiline
't'           Comment.Multiline
'h'           Comment.Multiline
'a'           Comment.Multiline
't'           Comment.Multiline
' '           Comment.Multiline
'n'           Comment.Multiline
'-'           Comment.Multiline
'l'           Comment.Multiline
'e'           Comment.Multiline
'n'           Comment.Multiline
'g'           Comment.Multiline
't'           Comment.Multiline
'h'           Comment.Multiline
'('           Comment.Multiline
'p'           Comment.Multiline
')'           Comment.Multiline
' '           Comment.Multiline
'i'           Comment.Multiline
's'           Comment.Multiline
' '           Comment.Multiline
'o'           Comment.Multiline
'd'           Comment.Multiline
'd'           Comment.Multiline
'.'           Comment.Multiline
'\n'          Comment.Multiline

'-'           Comment.Multiline
'-'           Comment.Multiline
'*/'          Comment.Multiline
'\n'          Text

'\n'          Text

' '           Text
' '           Text
' '           Text
' '           Text
'cycleTypes'  Name.Variable
' '           Text
':'           Operator
'='           Operator
'\n'          Text

' '           Text
' '           Text
' '           Text
' '           Text
'proc'        Keyword
'('           Punctuation
')'           Punctuation
'\n'          Text

'\t'          Text
'option'      Keyword
' '           Text
'remember'    Name.Variable
';'           Operator
'\n'          Text

'\t'          Text
'local'       Keyword
' '           Text
't'           Name.Variable
','           Operator
' '           Text
'p'           Name.Variable
','           Operator
' '           Text
'gen'         Name.Variable
';'           Operator
'\n'          Text

' '           Text
' '           Text
' '           Text
' '           Text
'begin'       Keyword
'\n'          Text

'\t'          Text
'userinfo'    Name.Function
'('           Punctuation
'3'           Literal.Number
','           Operator
' '           Text
'"cycleTypes: starting computation"' Literal.String
')'           Punctuation
';'           Operator
'\n'          Text

'\t'          Text
't'           Name.Variable
':'           Operator
'='           Operator
'table'       Name.Function
'('           Punctuation
')'           Punctuation
';'           Operator
'\n'          Text

'\n'          Text

'\t'          Text
'gen'         Name.Variable
' '           Text
':'           Operator
'='           Operator
' '           Text
'combinat::partitions::generator' Name.Function
'('           Punctuation
'n'           Name.Variable
')'           Punctuation
';'           Operator
'\n'          Text

'\t'          Text
'while'       Keyword
' '           Text
'('           Punctuation
'p'           Name.Variable
':'           Operator
'='           Operator
'gen'         Name.Function
'('           Punctuation
')'           Punctuation
')'           Punctuation
' '           Text
'<'           Operator
'>'           Operator
' '           Text
'FAIL'        Name.Constant
' '           Text
'do'          Keyword
'\n'          Text

'\t'          Text
' '           Text
' '           Text
' '           Text
' '           Text
'userinfo'    Name.Function
'('           Punctuation
'5'           Literal.Number
','           Operator
' '           Text
'"working on partition"' Literal.String
','           Operator
' '           Text
'p'           Name.Variable
')'           Punctuation
';'           Operator
'\n'          Text

'\t'          Text
' '           Text
' '           Text
' '           Text
' '           Text
'if'          Keyword
'('           Punctuation
'n'           Name.Variable
'-'           Operator
'nops'        Name.Function
'('           Punctuation
'p'           Name.Variable
')'           Punctuation
' '           Text
'mod'         Operator.Word
' '           Text
'2'           Literal.Number
'='           Operator
'0'           Literal.Number
')'           Punctuation
' '           Text
'then'        Keyword
'\n'          Text

'\t'          Text
'\t'          Text
'// Compute the size of the conjugacy class of Sn indexed by p' Comment.Single
'\n'          Text

'\t'          Text
'\t'          Text
'// and the cycle type of a permutation in this conjugacy class' Comment.Single
'\n'          Text

' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
't'           Name.Variable
'['           Punctuation
'combinat::partitions::toExp' Name.Function
'('           Punctuation
'p'           Name.Variable
','           Operator
'n'           Name.Variable
')'           Punctuation
']'           Punctuation
'\n'          Text

' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
':'           Operator
'='           Operator
' '           Text
'combinat::partitions::conjugacyClassSize' Name.Function
'('           Punctuation
'p'           Name.Variable
')'           Punctuation
';'           Operator
'\n'          Text

'\t'          Text
' '           Text
' '           Text
' '           Text
' '           Text
'end_if'      Keyword
';'           Operator
'\n'          Text

' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
' '           Text
'end_while'   Keyword
';'           Operator
'\n'          Text

'\t'          Text
't'           Name.Variable
';'           Operator
'\n'          Text

' '           Text
' '           Text
' '           Text
' '           Text
'end_proc'    Keyword
';'           Operator
'\n'          Text

'\n'          Text

'begin'       Keyword
'\n'          Text

' '           Text
' '           Text
' '           Text
' '           Text
'if'          Keyword
' '           Text
'testargs'    Name.Function
'('           Punctuation
')'           Punctuation
' '           Text
'then'        Keyword
'\n'          Text

'\t'          Text
'if'          Keyword
' '           Text
'args'        Name.Function
'('           Punctuation
'0'           Literal.Number
')'           Punctuation
' '           Text
'<'           Operator
'>'           Operator
' '           Text
'1'           Literal.Number
' '           Text
'then'        Keyword
' '           Text
'error'       Name.Function
'('           Punctuation
'"wrong no of args"' Literal.String
')'           Punctuation
';'           Operator
' '           Text
'end_if'      Keyword
';'           Operator
'\n'          Text

'\t'          Text
'if'          Keyword
' '           Text
'not'         Operator.Word
' '           Text
'testtype'    Name.Function
'('           Punctuation
'n'           Name.Variable
','           Operator
'DOM_INT'     Name.Class
')'           Punctuation
' '           Text
'then'        Keyword
'\n'          Text

'\t'          Text
' '           Text
' '           Text
' '           Text
' '           Text
'error'       Name.Function
'('           Punctuation
'"argument must be integer"' Literal.String
')'           Punctuation
'\n'          Text

'\t'          Text
'end_if'      Keyword
';'           Operator
'\n'          Text

'\t'          Text
'if'          Keyword
' '           Text
'n'           Name.Variable
' '           Text
'<'           Operator
' '           Text
'1'           Literal.Number
' '           Text
'then'        Keyword
'\n'          Text

'\t'          Text
' '           Text
' '           Text
' '           Text
' '           Text
'error'       Name.Function
'('           Punctuation
'"argument must be positive"' Literal.String
')'           Punctuation
'\n'          Text

'\t'          Text
'end_if'      Keyword
';'           Operator
'\n'          Text

' '           Text
' '           Text
' '           Text
' '           Text
'end_if'      Keyword
';'           Operator
'\n'          Text

'end_domain'  Keyword
':'           Operator
'\n'          Text
