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SandBoxSpeciesBayreuth2

Edit detail for SandBoxSpeciesBayreuth2 revision 5 of 10

	1 2 3 4 5 6 7 8 9 10
Editor: Thomas Feulner Time: 2008/06/17 00:48:45 GMT-7
Note:

added:
The identity in $S_5$ fixes every graph, so the coefficient of $x_1^5$ must be equal to coefficient(es, 5). This is the background to the equation
"es = cs(1,0,0,0...)"

\begin{axiom}
coefficient(es, 5)
\end{axiom}

Test the different Series defined for Species

axiom)cd ~/combinat/src
   The current FriCAS default directory is /var/lib/zope/combinat/src

axiom)re ../lib/combinat.input

axiom
)lib cscombinatversion.ao

axiom

Reading /var/lib/zope/combinat/src/cscombinatversion.asy
   LibraryInformationCombinat is now explicitly exposed in frame 
      initial 
   LibraryInformationCombinat will be automatically loaded when needed 
      from /var/lib/zope/combinat/src/cscombinatversion

axiom
)lib csaxcompat.ao

axiom

Reading /var/lib/zope/combinat/src/csaxcompat.asy
   ACCharacter is now explicitly exposed in frame initial 
   ACCharacter will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat
   ACFraction is now explicitly exposed in frame initial 
   ACFraction will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat
   ACIntegerTools is now explicitly exposed in frame initial 
   ACIntegerTools will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat
   ACInteger is now explicitly exposed in frame initial 
   ACInteger will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat
   ACList is now explicitly exposed in frame initial 
   ACList will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat
   ACMachineInteger is now explicitly exposed in frame initial 
   ACMachineInteger will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat
   ACPrimitiveArray is now explicitly exposed in frame initial 
   ACPrimitiveArray will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat
   ACString is now explicitly exposed in frame initial 
   ACString will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat
   ACSymbol is now explicitly exposed in frame initial 
   ACSymbol will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat
   Array is now explicitly exposed in frame initial 
   Array will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat
   ACLabelType is now explicitly exposed in frame initial 
   ACLabelType will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat
   OutputType is now explicitly exposed in frame initial 
   OutputType will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat
   TotallyOrderedType is now explicitly exposed in frame initial 
   TotallyOrderedType will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat

axiom
)lib csaxcompat2.ao

axiom

Reading /var/lib/zope/combinat/src/csaxcompat2.asy
   ExpressionTreeExpt is now explicitly exposed in frame initial 
   ExpressionTreeExpt will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat2
   ExpressionTreeLeaf is now explicitly exposed in frame initial 
   ExpressionTreeLeaf will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat2
   ExpressionTreePlus is now explicitly exposed in frame initial 
   ExpressionTreePlus will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat2
   ExpressionTreePrefix is now explicitly exposed in frame initial 
   ExpressionTreePrefix will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat2
   ExpressionTreeTimes is now explicitly exposed in frame initial 
   ExpressionTreeTimes will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat2
   ExpressionTree is now explicitly exposed in frame initial 
   ExpressionTree will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat2
   GeneratorException is now explicitly exposed in frame initial 
   GeneratorException will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat2
   Generator is now explicitly exposed in frame initial 
   Generator will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat2
   ExpressionTreeOperator is now explicitly exposed in frame initial 
   ExpressionTreeOperator will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat2
   ExpressionType is now explicitly exposed in frame initial 
   ExpressionType will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat2
   GeneratorExceptionType is now explicitly exposed in frame initial 
   GeneratorExceptionType will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csaxcompat2
   Partial is now explicitly exposed in frame initial 
   Partial will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat2
Won't parse: (Type)->NIL

axiom
)lib csistruc.ao

axiom

Reading /var/lib/zope/combinat/src/csistruc.asy
   SparseAdditiveArray is now explicitly exposed in frame initial 
   SparseAdditiveArray will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csistruc
   SparseFiniteMonoidRing is now explicitly exposed in frame initial 
   SparseFiniteMonoidRing will be automatically loaded when needed from
      /var/lib/zope/combinat/src/csistruc
   IndexedFreeAdditiveCombinationType is now explicitly exposed in 
      frame initial 
   IndexedFreeAdditiveCombinationType will be automatically loaded when
      needed from /var/lib/zope/combinat/src/csistruc
   IndexedFreeArithmeticType is now explicitly exposed in frame initial
   IndexedFreeArithmeticType will be automatically loaded when needed 
      from /var/lib/zope/combinat/src/csistruc

axiom
)lib csdistpoly.ao

axiom

Reading /var/lib/zope/combinat/src/csdistpoly.asy
   SparseDistributedPolynomial is now explicitly exposed in frame 
      initial 
   SparseDistributedPolynomial will be automatically loaded when needed
      from /var/lib/zope/combinat/src/csdistpoly

axiom
)lib csstream.ao

axiom

Reading /var/lib/zope/combinat/src/csstream.asy
   DataStream is now explicitly exposed in frame initial 
   DataStream will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csstream

axiom
)lib csseries.ao

axiom

Reading /var/lib/zope/combinat/src/csseries.asy
   SeriesOrder is now explicitly exposed in frame initial 
   SeriesOrder will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csseries
   FormalPowerSeries is now explicitly exposed in frame initial 
   FormalPowerSeries will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csseries
   FormalPowerSeriesCategory is now explicitly exposed in frame initial
   FormalPowerSeriesCategory will be automatically loaded when needed 
      from /var/lib/zope/combinat/src/csseries

axiom
)lib csidxpp.ao

axiom

Reading /var/lib/zope/combinat/src/csidxpp.asy
   SparseIndexedPowerProduct is now explicitly exposed in frame initial
   SparseIndexedPowerProduct will be automatically loaded when needed 
      from /var/lib/zope/combinat/src/csidxpp

axiom
)lib cssiprimes.ao

axiom

Reading /var/lib/zope/combinat/src/cssiprimes.asy
   SmallIntegerPrimes is now explicitly exposed in frame initial 
   SmallIntegerPrimes will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/cssiprimes

axiom
)lib cssitools.ao

axiom

Reading /var/lib/zope/combinat/src/cssitools.asy
   SmallIntegerTools is now explicitly exposed in frame initial 
   SmallIntegerTools will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/cssitools

axiom
)lib csgseries.ao

axiom

Reading /var/lib/zope/combinat/src/csgseries.asy
   CycleIndexSeries is now explicitly exposed in frame initial 
   CycleIndexSeries will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csgseries
   CycleIndexVariable is now explicitly exposed in frame initial 
   CycleIndexVariable will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csgseries
   ExponentialGeneratingSeries is now explicitly exposed in frame 
      initial 
   ExponentialGeneratingSeries will be automatically loaded when needed
      from /var/lib/zope/combinat/src/csgseries
   OrdinaryGeneratingSeries is now explicitly exposed in frame initial 
   OrdinaryGeneratingSeries will be automatically loaded when needed 
      from /var/lib/zope/combinat/src/csgseries

axiom
)lib csmultinom.ao

axiom

Reading /var/lib/zope/combinat/src/csmultinom.asy
   MultinomialTools is now explicitly exposed in frame initial 
   MultinomialTools will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csmultinom

axiom
)lib csspexpr.ao

axiom

Reading /var/lib/zope/combinat/src/csspexpr.asy
   SpeciesExpression is now explicitly exposed in frame initial 
   SpeciesExpression will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspexpr

axiom
)lib csspecies.ao

axiom

Reading /var/lib/zope/combinat/src/csspecies.asy
   RestrictedSpecies is now explicitly exposed in frame initial 
   RestrictedSpecies will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   SetSpecies is now explicitly exposed in frame initial 
   SetSpecies will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   SingletonSpecies is now explicitly exposed in frame initial 
   SingletonSpecies will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   ACIsomorphismType is now explicitly exposed in frame initial 
   ACIsomorphismType will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   Augment is now explicitly exposed in frame initial 
   Augment will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   Subset is now explicitly exposed in frame initial 
   Subset will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   CharacteristicSpecies is now explicitly exposed in frame initial 
   CharacteristicSpecies will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   Times is now explicitly exposed in frame initial 
   Times will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   Combination is now explicitly exposed in frame initial 
   Combination will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   Compose is now explicitly exposed in frame initial 
   Compose will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   Derivative is now explicitly exposed in frame initial 
   Derivative will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   DropEmpty is now explicitly exposed in frame initial 
   DropEmpty will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   EmptySetSpecies is now explicitly exposed in frame initial 
   EmptySetSpecies will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   FunctorialCompose is now explicitly exposed in frame initial 
   FunctorialCompose will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   MultiCompose is now explicitly exposed in frame initial 
   MultiCompose will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   MultiDerivative is now explicitly exposed in frame initial 
   MultiDerivative will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   MultiPlus is now explicitly exposed in frame initial 
   MultiPlus will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   MultiSetCombination is now explicitly exposed in frame initial 
   MultiSetCombination will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   MultiSetPartition is now explicitly exposed in frame initial 
   MultiSetPartition will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   MultiSet is now explicitly exposed in frame initial 
   MultiSet will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   MultiSubset is now explicitly exposed in frame initial 
   MultiSubset will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   CombinatorialSpecies is now explicitly exposed in frame initial 
   CombinatorialSpecies will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   MultiTimes is now explicitly exposed in frame initial 
   MultiTimes will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   Multiple is now explicitly exposed in frame initial 
   Multiple will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   NonEmpty is now explicitly exposed in frame initial 
   NonEmpty will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   IsomorphismTypeCategory is now explicitly exposed in frame initial 
   IsomorphismTypeCategory will be automatically loaded when needed 
      from /var/lib/zope/combinat/src/csspecies
   Partition is now explicitly exposed in frame initial 
   Partition will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   Plus is now explicitly exposed in frame initial 
   Plus will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies

axiom
)lib csexamples.ao

axiom

Reading /var/lib/zope/combinat/src/csexamples.asy
   ACBinaryTree is now explicitly exposed in frame initial 
   ACBinaryTree will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csexamples
   SetSpecies is already explicitly exposed in frame initial 
   SetSpecies will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csexamples
   Combination is already explicitly exposed in frame initial 
   Combination will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csexamples
   Generator is already explicitly exposed in frame initial 
   Generator will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csexamples
Won't parse: (ACLabelType)->NIL
Won't parse: (Type)->NIL

axiom
)lib csinterp.ao

axiom

Reading /var/lib/zope/combinat/src/csinterp.asy
   Interpret is now explicitly exposed in frame initial 
   Interpret will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csinterp
   InterpretingTools is now explicitly exposed in frame initial 
   InterpretingTools will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csinterp
   LabelSpecies is now explicitly exposed in frame initial 
   LabelSpecies will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csinterp

axiom
)lib csparse.ao

axiom

Reading /var/lib/zope/combinat/src/csparse.asy
   MyParser is now explicitly exposed in frame initial 
   MyParser will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csparse

A Graph Example:

aldor#includeDir "/var/lib/zope/combinat/include"
#libraryDir "/var/lib/zope/combinat/lib"
#include "combinat"
macro {
        E == EmptySetSpecies;
        X == SingletonSpecies;
        + == Plus;
        * == Times;
}
import from ACInteger;
TwoSet(L: ACLabelType): CombinatorialSpecies L == RestrictedSpecies(SetSpecies, 2)(L) add;
Graph(L: ACLabelType): CombinatorialSpecies L == FunctorialCompose(Subset, SetSpecies*TwoSet)(L)

aldor

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/3184588646873157811-25px002.as using 
      AXIOM-XL compiler and options 
-O -Fasy -Fao -Flsp -laxiom -Mno-AXL_W_WillObsolete -DAxiom -Y $AXIOM/algebra
      Use the system command )set compiler args to change these 
      options.
#1 (Warning) Deprecated message prefix: use `ALDOR_' instead of `_AXL'
"/var/zope2/var/LatexWiki/3184588646873157811-25px002.as", line 13: 
Graph(L: ACLabelType): CombinatorialSpecies L == FunctorialCompose(Subset, SetSpecies*TwoSet)(L)
......^
[L13 C7] #2 (Warning) Function returns a domain that might not be constant (which may cause problems if it is used in a dependent type).
   Compiling Lisp source code from file 
      ./3184588646873157811-25px002.lsp
   Issuing )library command for 3184588646873157811-25px002
   Reading /var/lib/zope/combinat/src/3184588646873157811-25px002.asy
   TwoSet is now explicitly exposed in frame initial 
   TwoSet will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/3184588646873157811-25px002
   Graph is now explicitly exposed in frame initial 
   Graph will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/3184588646873157811-25px002

axiom
Z := ACInteger;

Type: Domain

axiom
X:=TwoSet(Z);

Type: Domain

axiom[structures(labels)$X]$ACLIST(X)
   There are 1 exposed and 0 unexposed library operations named
      structures having 1 argument(s) but none was determined to be
      applicable. Use HyperDoc Browse, or issue
                           )display op structures
      to learn more about the available operations. Perhaps
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.
   Cannot find a definition or applicable library operation named
      structures with argument type(s)
                               Variable labels
      Perhaps you should use "@" to indicate the required return type,
      or "$" to specify which version of the function you need.
labels3: SetSpecies Z := set [10::Z,20::Z]

(1)

Type: SetSpecies? ACInteger?

axiom
[structures(labels3)$X]$ACLIST(X)

(2)

Type: ACList? TwoSet? ACInteger?

axiom
es: ExponentialGeneratingSeries := generatingSeries()$Graph(Z);

Type: ExponentialGeneratingSeries?

axiom
[coefficient(es, i) for i in 0..5]

(3)

Type: List ACFraction? ACInteger?

axiom
os: OrdinaryGeneratingSeries := isomorphismTypeGeneratingSeries()$Graph(Z);

Type: OrdinaryGeneratingSeries?

axiom
[coefficient(os, i) for i in 0..5]

(4)

Type: List ACInteger?

Some problems are appearing, calling [structures(labels3)ACLIST(Graph(Z))

And what is the number of Isomorphism Types

axiom
cs: CycleIndexSeries := cycleIndexSeries()$Graph(Z);

Type: CycleIndexSeries?

axiom
coefficient(cs, 5)

(5)

Type: SparseDistributedPolynomial?(ACFraction? ACInteger?,CycleIndexVariable?,SparseIndexedPowerProduct?(CycleIndexVariable?,ACMachineInteger?))

The identity in fixes every graph, so the coefficient of must be equal to coefficient(es, 5). This is the background to the equation "es = cs(1,0,0,0...)"

axiom
coefficient(es, 5)

(6)

Type: ACFraction? ACInteger?