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)abbrev package CYCLES CycleIndicators
++ Polya-Redfield enumeration by cycle indices.
++ Author: William H. Burge
++ Date Created: 1986
++ Date Last Updated: 11 Feb 1992
++ Keywords:Polya, Redfield, enumeration
++ Examples:
++ References: J.H.Redfield, 'The Theory of Group-Reduced Distributions',
++ American J. Math., 49 (1927) 433-455.
++ G.Polya, 'Kombinatorische Anzahlbestimmungen fur Gruppen,
++ Graphen und chemische Verbindungen', Acta Math. 68
++ (1937) 145-254.
++ Description: Enumeration by cycle indices.
CycleIndicators: Exports == Implementation where
I ==> Integer
L ==> List
B ==> Boolean
SPOL ==> SymmetricPolynomial
PTN ==> Partition
RN ==> Fraction Integer
FR ==> Factored Integer
macro NNI == NonNegativeInteger
macro PI == PositiveInteger
Exports ==> with
complete: PI -> SPOL RN
++\spad{complete n} is the \spad{n} th complete homogeneous
++ symmetric function expressed in terms of power sums.
++ Alternatively it is the cycle index of the symmetric
++ group of degree n.
powerSum: PI -> SPOL RN
++\spad{powerSum n} is the \spad{n} th power sum symmetric
++ function.
elementary: PI -> SPOL RN
++\spad{elementary n} is the \spad{n} th elementary symmetric
++ function expressed in terms of power sums.
alternating: PI -> SPOL RN
++\spad{alternating n} is the cycle index of the
++ alternating group of degree n.
cyclic: PI -> SPOL RN --cyclic group
++\spad{cyclic n} is the cycle index of the
++ cyclic group of degree n.
dihedral: PI -> SPOL RN --dihedral group
++\spad{dihedral n} is the cycle index of the
++ dihedral group of degree n.
graphs: PI -> SPOL RN
++\spad{graphs n} is the cycle index of the group induced on
++ the edges of a graph by applying the symmetric function to the
++ n nodes.
cap: (SPOL RN,SPOL RN) -> RN
++\spad{cap(s1,s2)}, introduced by Redfield,
++ is the scalar product of two cycle indices.
cup: (SPOL RN,SPOL RN) -> SPOL RN
++\spad{cup(s1,s2)}, introduced by Redfield,
++ is the scalar product of two cycle indices, in which the
++ power sums are retained to produce a cycle index.
eval: SPOL RN -> RN
++\spad{eval s} is the sum of the coefficients of a cycle index.
wreath: (SPOL RN,SPOL RN) -> SPOL RN
++\spad{wreath(s1,s2)} is the cycle index of the wreath product
++ of the two groups whose cycle indices are \spad{s1} and
++ \spad{s2}.
SFunction:L PI -> SPOL RN
++\spad{SFunction(li)} is the S-function of the partition \spad{li}
++ expressed in terms of power sum symmetric functions.
skewSFunction:(L I,L I) -> SPOL RN
++\spad{skewSFunction(li1,li2)} is the S-function
++ of the partition difference \spad{li1 - li2}
++ expressed in terms of power sum symmetric functions.
Implementation ==> add
import IntegerNumberTheoryFunctions
import Partition
trm: PTN -> SPOL RN
trm pt == monomial(inv(pdct(pt) :: RN),pt)
list: Stream PTN -> L PTN
list st == entries complete st
complete i ==
i=0 => 1
+/[trm pt for pt in list partitions i]
even?: PTN -> B
even? p == even?( #([i for i in parts p | even? i]))
alternating i ==
2 * _+/[trm p for p in list partitions i | even? p]
elementary i ==
i=0 => 1
+/[(spol := trm pt; even? pt => spol; -spol)
for pt in list partitions i]
divisors: I -> L I
divisors n ==
b := factors(n :: FR)
c := concat(1,"append"/
[[a.factor**j for j in 1..a.exponent] for a in b]);
if #(b) = 1 then c else concat(n,c)
ss: (PI,I) -> SPOL RN
ss(n,m) ==
li : L PI := [n for j in 1..m]
monomial(1,partition li)
powerSum n == ss(n,1)
cyclic n ==
n = 1 => powerSum 1
+/[(eulerPhi(i) / n) * ss(i::PI,numer(n/i)) for i in divisors n]
dihedral n ==
k := n quo 2
odd? n => (1/2) * cyclic n + (1/2) * ss(2,k) * powerSum 1
(1/2) * cyclic n + (1/4) * ss(2,k) + (1/4) * ss(2,k-1) * ss(1,2)
trm2: PTN -> SPOL RN
trm2 li ==
lli := powers( li)$PTN
xx := 1/(pdct li)
prod : SPOL RN := 1
for ll in lli repeat
ll0 := first ll; ll1 := second ll
k := ll0 quo 2
c :=
odd? ll0 => ss(ll0,ll1 * k)
ss(k::PI,ll1) * ss(ll0,ll1 * (k - 1))
c := c * ss(ll0,ll0 * ((ll1*(ll1 - 1)) quo 2))
prod2 : SPOL RN := 1
for r in lli | first(r) < ll0 repeat
r0 := first r; r1 := second r
prod2 := ss(lcm(r0,ll0)::PI,gcd(r0,ll0) * r1 * ll1) * prod2
prod := c * prod2 * prod
xx * prod
graphs n == +/[trm2 p for p in list(partitions n)]
cupp: (PTN,SPOL RN) -> SPOL RN
cupp(pt,spol) ==
zero? spol => 0
(dg := degree spol) < pt => 0
dg = pt => (pdct pt) * monomial(leadingCoefficient spol,dg)
cupp(pt,reductum spol)
cup(spol1,spol2) ==
zero? spol1 => 0
p := leadingCoefficient(spol1) * cupp(degree spol1,spol2)
p + cup(reductum spol1,spol2)
eval spol ==
zero? spol => 0
leadingCoefficient(spol) + eval(reductum spol)
cap(spol1,spol2) == eval cup(spol1,spol2)
mtpol: (PI,SPOL RN) -> SPOL RN
mtpol(n,spol)==
zero? spol => 0
deg := partition [n*k for k in (degree spol)::L(PI)]
monomial(leadingCoefficient spol,deg) + mtpol(n,reductum spol)
evspol: ((PI -> SPOL RN),SPOL RN) -> SPOL RN
evspol(fn2,spol) ==
zero? spol => 0
lc := leadingCoefficient spol
prod := */[fn2 i for i in (degree spol)::L(PI)]
lc * prod + evspol(fn2,reductum spol)
wreath(spol1,spol2) == evspol(mtpol(#1,spol2),spol1)
SFunction li==
a:Matrix SPOL RN :=
matrix [[complete((k -j+i)::PI) for k in li for j in 1..#li]
for i in 1..#li]
determinant a
roundup:(L I,L I)-> L I
roundup(li1,li2)==
#li1 > #li2 => roundup(li1,concat(li2,0))
li2
skewSFunction(li1,li2)==
#li1 < #li2 =>
error "skewSFunction: partition1 does not include partition2"
li2:=roundup (li1,li2)
a:Matrix SPOL RN:=matrix [[complete((k-li2.i-j+i)::PI)
for k in li1 for j in 1..#li1] for i in 1..#li1]
determinant a
)abbrev package EVALCYC EvaluateCycleIndicators
++ Author: William H. Burge
++ Date Created: 1986
++ Date Last Updated: Feb 1992
++ Basic Operations:
++ Related Domains:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ Examples:
++ References:
++ Description: This package is to be used in conjuction with
++ the CycleIndicators package. It provides an evaluation
++ function for SymmetricPolynomials.
EvaluateCycleIndicators(F):T==C where
F:Algebra Fraction Integer
I==>Integer
L==>List
SPOL==SymmetricPolynomial
RN==>Fraction Integer
PR==>Polynomial(RN)
PTN==>Partition()
lc ==> leadingCoefficient
red ==> reductum
T== with
eval:((I->F),SPOL RN)->F
++\spad{eval(f,s)} evaluates the cycle index s by applying
++ the function f to each integer in a monomial partition,
++ forms their product and sums the results over all monomials.
C== add
evp:((I->F),PTN)->F
fn:I->F
pt:PTN
spol:SPOL RN
evp(fn, pt)== */[fn i for i in pt::L(PositiveInteger)]
eval(fn,spol)==
if spol=0
then 0
else ((lc spol)* evp(fn,degree spol)) + eval(fn,red spol)
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