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Test species in axiom/FriCAS.

The species project is written in Aldor. Thus we need to load the libraries explicitly:

axiom
)cd /var/lib/zope/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
   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
   ACMachineInteger is now explicitly exposed in frame initial 
   ACMachineInteger 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
   ACCharacter is now explicitly exposed in frame initial 
   ACCharacter 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
   ACInteger is now explicitly exposed in frame initial 
   ACInteger 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
   ACList is now explicitly exposed in frame initial 
   ACList 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
   ACPrimitiveArray is now explicitly exposed in frame initial 
   ACPrimitiveArray 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
   ACFraction is now explicitly exposed in frame initial 
   ACFraction 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
   Generator is now explicitly exposed in frame initial 
   Generator 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
   GeneratorExceptionType is now explicitly exposed in frame initial 
   GeneratorExceptionType 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
   ExpressionType is now explicitly exposed in frame initial 
   ExpressionType 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
   ExpressionTreeOperator is now explicitly exposed in frame initial 
   ExpressionTreeOperator 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
   ExpressionTreeTimes is now explicitly exposed in frame initial 
   ExpressionTreeTimes will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csaxcompat2
   ExpressionTreeExpt is now explicitly exposed in frame initial 
   ExpressionTreeExpt 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
   ExpressionTreeLeaf is now explicitly exposed in frame initial 
   ExpressionTreeLeaf 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
   IndexedFreeAdditiveCombinationType is now explicitly exposed in 
      frame initial 
   IndexedFreeAdditiveCombinationType will be automatically loaded when
      needed from /var/lib/zope/combinat/src/csistruc
   SparseAdditiveArray is now explicitly exposed in frame initial 
   SparseAdditiveArray 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 SparseFiniteMonoidRing is now explicitly exposed in frame initial SparseFiniteMonoidRing 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
   FormalPowerSeriesCategory is now explicitly exposed in frame initial
FormalPowerSeriesCategory will be automatically loaded when needed from /var/lib/zope/combinat/src/csseries 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
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
   OrdinaryGeneratingSeries is now explicitly exposed in frame initial 
   OrdinaryGeneratingSeries 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
   CycleIndexVariable is now explicitly exposed in frame initial 
   CycleIndexVariable will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csgseries
   CycleIndexSeries is now explicitly exposed in frame initial 
   CycleIndexSeries 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
   Multiple is now explicitly exposed in frame initial 
   Multiple 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
   ACIsomorphismType is now explicitly exposed in frame initial 
   ACIsomorphismType 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
   MultiSet is now explicitly exposed in frame initial 
   MultiSet 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
   CharacteristicSpecies is now explicitly exposed in frame initial 
   CharacteristicSpecies 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
   SingletonSpecies is now explicitly exposed in frame initial 
   SingletonSpecies will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/csspecies
   RestrictedSpecies is now explicitly exposed in frame initial 
   RestrictedSpecies 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
   Partition is now explicitly exposed in frame initial 
   Partition 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
   MultiSetCombination is now explicitly exposed in frame initial 
   MultiSetCombination 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
   Subset is now explicitly exposed in frame initial 
   Subset 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
   MultiPlus is now explicitly exposed in frame initial 
   MultiPlus 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
   MultiTimes is now explicitly exposed in frame initial 
   MultiTimes 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
   MultiCompose is now explicitly exposed in frame initial 
   MultiCompose 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
   MultiDerivative is now explicitly exposed in frame initial 
   MultiDerivative 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
   Derivative is now explicitly exposed in frame initial 
   Derivative 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
axiom
)lib csexamples.ao
axiom
Reading /var/lib/zope/combinat/src/csexamples.asy
   SetSpecies is already explicitly exposed in frame initial 
   SetSpecies 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
   ACBinaryTree is now explicitly exposed in frame initial 
   ACBinaryTree 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
Won't parse: (ACLabelType)->NIL
Won't parse: (Type)->NIL
axiom
)lib csinterp.ao
axiom
Reading /var/lib/zope/combinat/src/csinterp.asy
   LabelSpecies is now explicitly exposed in frame initial 
   LabelSpecies 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
   Interpret is now explicitly exposed in frame initial 
   Interpret 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

Let's define a binary tree.

aldor
#includeDir "/var/lib/zope/combinat/include"
#libraryDir "/var/lib/zope/combinat/lib"
#include "combinat"
macro { E == EmptySetSpecies; X == SingletonSpecies; + == Plus; * == Times; }
A(L: LabelType): CombinatorialSpecies L == (E + X*A*A)(L) add;
aldor
   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/5200085684951427974-25px002.as using 
      AXIOM-XL compiler and options 
-O -Fasy -Fao -Flsp -laxiom -Mno-ALDOR_W_WillObsolete -DAxiom -Y $AXIOM/algebra -I $AXIOM/algebra
      Use the system command )set compiler args to change these 
      options.
   Compiling Lisp source code from file 
      ./5200085684951427974-25px002.lsp
   Issuing )library command for 5200085684951427974-25px002
   Reading /var/lib/zope/combinat/src/5200085684951427974-25px002.asy
   A is now explicitly exposed in frame initial 
   A will be automatically loaded when needed from 
      /var/lib/zope/combinat/src/5200085684951427974-25px002

aldor
#includeDir "/var/lib/zope/combinat/include"
#libraryDir "/var/lib/zope/combinat/lib"
#assert MacrosCombinat
#assert Axiom
#include "combinat"
macro {
        SPECIES == (L: LabelType) -> CombinatorialSpecies L;
        V == CycleIndexVariable;
        NonNegativeMachineInteger == I;
        T == SparseIndexedPowerProduct(V, NonNegativeMachineInteger);
        P == SparseDistributedPolynomial(Q, V, T);
}
LinearOrder(L: LabelType): with {
CombinatorialSpecies L;
coerce: % -> List L;
} == List L add {
        Rep == List L;
        import from Rep;
coerce(x: %): List L == rep x; local lists(l: List L): Generator List L == generate { empty? l => yield l; current := l; c := first current; for u in lists(rest l) repeat yield cons(c, u); assert(not empty? current); while not empty?(tmp := rest current) repeat { c := first tmp; setrest!(current, rest tmp); -- remove c from l for u in lists l repeat yield cons(c, u); setrest!(current, tmp); -- put c back into l current := tmp; } } structures(s: SetSpecies L): Generator % == generate { for l in lists(s :: List L) repeat yield per l; } local LinearOrderIsomorphismType: IsomorphismTypeCategory L == add { isomorphismTypes(s: MultiSet L): Generator % == never; (x:%) = (y:%): Boolean == never; (tw: TextWriter) << (x: %): TextWriter == never; }
IsomorphismType: IsomorphismTypeCategory L == LinearOrderIsomorphismType; generatingSeries: ExponentialGeneratingSeries == { (stream(1$Q)$DataStream(Q)) :: ExponentialGeneratingSeries; } isomorphismTypeGeneratingSeries: OrdinaryGeneratingSeries == { (stream(1$Z)$DataStream(Z)) :: OrdinaryGeneratingSeries; } local cisGenerator: Generator P == generate { import from I, T, P; x1: V := 1::V; for n: I in 0.. repeat yield power(x1, n) :: P; } cycleIndexSeries: CycleIndexSeries == cisGenerator :: CycleIndexSeries; import from String; expression: SpeciesExpression == leaf("LinearOrder"); }
Cycle(L: LabelType): with { CombinatorialSpecies L; coerce: % -> List L; cycle: List L -> %; } == List L add { Rep == List L; import from I, Rep;
local cisCycle(ao: I): Generator P == generate { macro PrimePowerProduct == SparseIndexedPowerProduct(I, I);
local multiply(k: PrimePowerProduct): I == { r: I := 1; for ep in k repeat {(e, p) := ep; r := r * p^e} r; }
local eulerPhi(t: SparseIndexedPowerProduct(I, I)): I == { phi: I := 1; for ep in t repeat { (e, p) := ep; phi := phi * p^(e-1) * (p-1) } phi; }
local cisCoefficient(n: I): P == BugWorkaround( PrimePowerProduct has with { divisors: % -> Generator %; /: (%, %) -> %; } ){ import from Z, V, SmallIntegerTools; nn: PrimePowerProduct := factor n; p: P := 0; for m in divisors nn repeat { k: PrimePowerProduct := nn/m; q: Q := (eulerPhi(k) :: Z) / (n :: Z); xk: V := multiply(k) :: V; t: T := power(xk, multiply m); p := [q, t]$P + p; } p; } yield 0$P; for n:I in 1.. repeat yield cisCoefficient(n); } coerce(x: %): List L == rep x; cycle(l: List L): % == per l; structures(s: SetSpecies L): Generator % == generate { import from LinearOrder L; if not empty? s then { l: List L := s :: List L; u := first l; for t in structures(set rest l)$LinearOrder(L) repeat { yield per cons(u, t :: List L); } } }
local CycleIsomorphismType: IsomorphismTypeCategory L == add { isomorphismTypes(s: MultiSet L): Generator % == never; (x:%) = (y:%): Boolean == never; (tw: TextWriter) << (x: %): TextWriter == never; } IsomorphismType: IsomorphismTypeCategory L == CycleIsomorphismType; local cycleOrder(): SeriesOrder == 1 :: SeriesOrder; egsCycle(ao: I): Generator Q == generate { import from Z, Q; yield 0; for n:I in 1.. repeat yield inv(n :: Z); } generatingSeries: ExponentialGeneratingSeries == new(egsCycle, cycleOrder); ogsCycle(ao: I): Generator Z == generate {yield 0$Z; yield 1$Z}; isomorphismTypeGeneratingSeries: OrdinaryGeneratingSeries == { new(ogsCycle, cycleOrder); } cycleIndexSeries: CycleIndexSeries == new(cisCycle, cycleOrder); import from String; expression: SpeciesExpression == leaf("Cycle"); }
aldor
   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/6373670188426008557-25px003.as using 
      AXIOM-XL compiler and options 
-O -Fasy -Fao -Flsp -laxiom -Mno-ALDOR_W_WillObsolete -DAxiom -Y $AXIOM/algebra -I $AXIOM/algebra
      Use the system command )set compiler args to change these 
      options.
"/var/zope2/var/LatexWiki/6373670188426008557-25px003.as", line 94: 
){
.^
[L94 C2] #1 (Warning) Suspicious juxtaposition.  Check for missing `;'.
Check indentation if you are using `#pile'.
Compiling Lisp source code from file ./6373670188426008557-25px003.lsp Issuing )library command for 6373670188426008557-25px003 Reading /var/lib/zope/combinat/src/6373670188426008557-25px003.asy LinearOrder is now explicitly exposed in frame initial LinearOrder will be automatically loaded when needed from /var/lib/zope/combinat/src/6373670188426008557-25px003 Cycle is now explicitly exposed in frame initial Cycle will be automatically loaded when needed from /var/lib/zope/combinat/src/6373670188426008557-25px003

axiom
labels: SetSpecies ACINT := set [i::ACINT for i in 1..3]

\label{eq1}\left[ 1, \: 2, \: 3 \right](1)
Type: SetSpecies?(ACInteger?)
axiom
)set output tex off
 
axiom
)set output algebra on
[structures(labels)$Cycle(ACINT)]$ACLIST(Cycle(ACINT))
(2) [[1,2,3],[1,3,2]]
Type: ACList?(Cycle(ACInteger?))

axiom
I := ACMachineInteger
(3) ACMachineInteger
Type: Type
axiom
Z := ACInteger
(4) ACInteger
Type: Type
axiom
Q := ACFraction Z
(5) ACFraction(ACInteger)
Type: Type
axiom
V := CycleIndexVariable
(6) CycleIndexVariable
Type: Type
axiom
T := SparseIndexedPowerProduct(V, I)
(7) SparseIndexedPowerProduct(CycleIndexVariable,ACMachineInteger)
Type: Type
axiom
P := SparseDistributedPolynomial(Q, V, T)
(8) SparseDistributedPolynomial(ACFraction(ACInteger),CycleIndexVariable,SparseIn dexedPowerProduct(CycleIndexVariable,ACMachineInteger))
Type: Type
axiom
S := CycleIndexSeries;
Type: Type
axiom
X := A(Z)
(10) A(ACInteger)
Type: Type
axiom
)show X
A(ACInteger) is a domain constructor. Abbreviation for A is A This constructor is exposed in this frame. ------------------------------- Operations --------------------------------
?=? : (%,%) -> Boolean coerce : % -> OutputForm hash : % -> SingleInteger latex : % -> String ?~=? : (%,%) -> Boolean ?<<? : (OutputForm,%) -> OutputForm IsomorphismType : () -> IsomorphismTypeCategory(ACInteger) cycleIndexSeries : () -> CycleIndexSeries expression : () -> SpeciesExpression generatingSeries : () -> ExponentialGeneratingSeries isomorphismTypeGeneratingSeries : () -> OrdinaryGeneratingSeries structures : SetSpecies(ACInteger) -> Generator(%)
labels: SetSpecies Z := set [i::Z for i in 1..3]
(11) [1,2,3]
Type: SetSpecies?(ACInteger?)
axiom
)set output tex off
 
axiom
)set output algebra on
[structures(labels)$X]$ACLIST(X)
(12) [((1, "nil"), ((2, "nil"), ((3, "nil"), "nil"))), ((1, "nil"), ((3, "nil"), ((2, "nil"), "nil"))), ((1, "nil"), ((2, ((3, "nil"), "nil")), "nil")), ((1, "nil"), ((3, ((2, "nil"), "nil")), "nil")), ((2, "nil"), ((1, "nil"), ((3, "nil"), "nil"))), ((2, "nil"), ((3, "nil"), ((1, "nil"), "nil"))), ((2, "nil"), ((1, ((3, "nil"), "nil")), "nil")), ((2, "nil"), ((3, ((1, "nil"), "nil")), "nil")), ((1, ((2, "nil"), "nil")), ((3, "nil"), "nil")), ((2, ((1, "nil"), "nil")), ((3, "nil"), "nil")), ((3, "nil"), ((1, "nil"), ((2, "nil"), "nil"))), ((3, "nil"), ((2, "nil"), ((1, "nil"), "nil"))), ((3, "nil"), ((1, ((2, "nil"), "nil")), "nil")), ((3, "nil"), ((2, ((1, "nil"), "nil")), "nil")), ((1, ((3, "nil"), "nil")), ((2, "nil"), "nil")), ((3, ((1, "nil"), "nil")), ((2, "nil"), "nil")), ((2, ((3, "nil"), "nil")), ((1, "nil"), "nil")), ((3, ((2, "nil"), "nil")), ((1, "nil"), "nil")), ((1, ((2, "nil"), ((3, "nil"), "nil"))), "nil"), ((1, ((3, "nil"), ((2, "nil"), "nil"))), "nil"), ((1, ((2, ((3, "nil"), "nil")), "nil")), "nil"), ((1, ((3, ((2, "nil"), "nil")), "nil")), "nil"), ((2, ((1, "nil"), ((3, "nil"), "nil"))), "nil"), ((2, ((3, "nil"), ((1, "nil"), "nil"))), "nil"), ((2, ((1, ((3, "nil"), "nil")), "nil")), "nil"), ((2, ((3, ((1, "nil"), "nil")), "nil")), "nil"), ((3, ((1, "nil"), ((2, "nil"), "nil"))), "nil"), ((3, ((2, "nil"), ((1, "nil"), "nil"))), "nil"), ((3, ((1, ((2, "nil"), "nil")), "nil")), "nil"), ((3, ((2, ((1, "nil"), "nil")), "nil")), "nil")]
Type: ACList?(A(ACInteger?))
axiom
)set output tex on
 
axiom
)set output algebra off

Let's count how many structures of a certain size exist. This is encoded in the generating series. Note that this is an exponential generating series.

axiom
es: ExponentialGeneratingSeries := generatingSeries()$X;
Type: ExponentialGeneratingSeries?
axiom
[coefficient(es, i) for i in 0..10]

\label{eq2}\left[ 1, \: 1, \: 2, \: 5, \:{14}, \:{42}, \:{132}, \:{429}, \:{1430}, \:{4862}, \:{16796}\right](2)
Type: List(ACFraction?(ACInteger?))
axiom
[count(es, i)       for i in 0..10]

\label{eq3}\begin{array}{@{}l}
\displaystyle
\left[ 1, \: 1, \: 4, \:{30}, \:{336}, \:{5040}, \:{95040}, \:{2
162160}, \:{57657600}, \:{1764322560}, \: \right.
\
\
\displaystyle
\left.{60949324800}\right] 
(3)
Type: List(ACInteger?)

AldorCombinat? can also count the number of isomorphism types of structures. That is encoded in the isomorphism type series.

axiom
os: OrdinaryGeneratingSeries := isomorphismTypeGeneratingSeries()$X;
Type: OrdinaryGeneratingSeries?
axiom
[coefficient(os, i) for i in 0..10]

\label{eq4}\left[ 1, \: 1, \: 2, \: 5, \:{14}, \:{42}, \:{132}, \:{429}, \:{1430}, \:{4862}, \:{16796}\right](4)
Type: List(ACInteger?)
axiom
[count(os, i)       for i in 0..10]

\label{eq5}\left[ 1, \: 1, \: 2, \: 5, \:{14}, \:{42}, \:{132}, \:{429}, \:{1430}, \:{4862}, \:{16796}\right](5)
Type: List(ACInteger?)
axiom
cs: S := cycleIndexSeries()$X;
Type: CycleIndexSeries?

The output of the next command apparently causes a "Proxy Error" on the mathaction server. This can be avoided by changing the output from LaTeX to ASCII mode:

axiom
)set output tex off
 
axiom
)set output algebra on
coeffs3: ACList P := [coefficient(cs, i) for i in 0..5]
(20) [(1)*1, (1)*x1^1, (2)*x1^2, (5)*x1^3, (14)*x1^4, (42)*x1^5]
Type: ACList?(SparseDistributedPolynomial?(ACFraction?(ACInteger?),CycleIndexVariable?,SparseIndexedPowerProduct?(CycleIndexVariable?,ACMachineInteger?)))
axiom
)set output tex on
 
axiom
)set output algebra off

P is the polynomial domain in infinitely many variables with rational coefficients.

axiom
p: P := 1$P

\label{eq6}1(6)
Type: SparseDistributedPolynomial?(ACFraction?(ACInteger?),CycleIndexVariable?,SparseIndexedPowerProduct?(CycleIndexVariable?,ACMachineInteger?))
axiom
v: V := 3::I::V

\label{eq7}x_{3}(7)
Type: CycleIndexVariable?
axiom
t: T := power(v,2) * power(5::I::V,3)

\label{eq8}{{x_{3}}^2}\ {{x_{5}}^3}(8)
Type: SparseIndexedPowerProduct?(CycleIndexVariable?,ACMachineInteger?)
axiom
t2: T := power(v,3) * power(1::I::V,2)

\label{eq9}{{x_{1}}^2}\ {{x_{3}}^3}(9)
Type: SparseIndexedPowerProduct?(CycleIndexVariable?,ACMachineInteger?)
axiom
p := t :: P + 3*t2

\label{eq10}{3 \ {{x_{1}}^2}\ {{x_{3}}^3}}+{{{x_{3}}^2}\ {{x_{5}}^3}}(10)
Type: SparseDistributedPolynomial?(ACFraction?(ACInteger?),CycleIndexVariable?,SparseIndexedPowerProduct?(CycleIndexVariable?,ACMachineInteger?))

Let's do some more advanced stuff with Aldor-Combinat . See the functorial composition testpage for more details.

Here we compute the cycle index series of simple graphs.

axiom
e: S := cycleIndexSeries() $ SetSpecies(Z);
Type: CycleIndexSeries?
axiom
e2: S := term(coefficient(e, 2), 2);
Type: CycleIndexSeries?
axiom
h: S := functorialCompose(e * e, e2 * e);
Type: CycleIndexSeries?
axiom
coefficient(h, 0)

\label{eq11}1(11)
Type: SparseDistributedPolynomial?(ACFraction?(ACInteger?),CycleIndexVariable?,SparseIndexedPowerProduct?(CycleIndexVariable?,ACMachineInteger?))
axiom
coefficient(h, 1)

\label{eq12}x_{1}(12)
Type: SparseDistributedPolynomial?(ACFraction?(ACInteger?),CycleIndexVariable?,SparseIndexedPowerProduct?(CycleIndexVariable?,ACMachineInteger?))
axiom
coefficient(h, 2)
>> System error: Couldn't load "/var/lib/zope/combinat/src/cssiprimes": file does not exist.

Suspicious juxtaposition --Bill Page, Tue, 17 Jun 2008 16:08:35 -0700 reply
What is the intention of BugWorkaround in this code:
  local cisCoefficient(n: I): P == BugWorkaround(
    PrimePowerProduct has with {
            divisors: % -> Generator %;
            /: (%, %) -> %;
    }
  ){
        import from Z, V, SmallIntegerTools;
        ...

It results in the message:

  L94 C2] #2 (Warning) Suspicious juxtaposition.  Check for missing `;'.




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