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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{src/algebra prtition.spad}
\author{William H. Burge}
\maketitle
\begin{abstract}
\end{abstract}
\tableofcontents
\eject
\section{domain PRTITION Partition}
<<domain PRTITION Partition>>=
import Integer
import List
)abbrev domain PRTITION Partition
++ Domain for partitions of positive integers
++ Author: William H. Burge
++ Date Created: 29 October 1987
++ Date Last Updated: April 17, 2010
++ Description:
++ Partition is an OrderedCancellationAbelianMonoid which is used
++ as the basis for symmetric polynomial representation of the
++ sums of powers in SymmetricPolynomial. Thus, \spad{(5 2 2 1)} will
++ represent \spad{s5 * s2**2 * s1}.
++ Keywords:
++ Examples:
++ References:
Partition(): Exports == Implementation where
macro L == List
macro I == Integer
macro PI == PositiveInteger
macro OUT == OutputForm
macro NNI == NonNegativeInteger
macro UN == Union(%,"failed")
Exports == Join(OrderedCancellationAbelianMonoid, CoercibleTo L PI) with
partition: L PI -> %
++ partition(li) converts a list of integers li to a partition
parts: % -> L PI
++ \spad{parts x} returns the list of decreasing integer sequence
++ making up the partition \spad{x}.
#: % -> NNI
++ \spad{#x} returns the sum of all parts of the partition \spad{x}.
partitions: NNI -> Stream %
++ \spad{partitions n} returns the stream of all partitions of
++ size \spad{n}.
powers: % -> List Pair(PI,PI)
++ powers(x) returns a list of pairs. The second component of
++ each pair is the multiplicity with which the first component
++ occurs in li.
pdct: % -> PI
++ \spad{pdct(a1**n1 a2**n2 ...)} returns
++ \spad{n1! * a1**n1 * n2! * a2**n2 * ...}.
++ This function is used in the package \spadtype{CycleIndicators}.
conjugate: % -> %
++ conjugate(p) returns the conjugate partition of a partition p
Implementation == add
Rep == L PI
0 == per nil
coerce(s: %): L PI == rep s
partition list == per sort(#2 < #1,list)
parts x == rep x
#x ==
empty? rep x => 0
+/[n for n in rep x]
allPartitions(n: NNI, k: NNI): Stream % ==
zero? n => cons(0,empty()$Stream(%))
zero? k => empty()$Stream(%)
one? k => cons(partition [1 for i in 1..n], empty()$Stream(%))
s :=
n < k => empty()$Stream(%)
allPartitions((n-k)::NNI,k)
concat(map(per(cons(k::PI, rep #1)),s), allPartitions(n,(k-1)::NNI))
partitions n == allPartitions(n,n)
zero? x == empty? rep x
x < y == rep x < rep y
x = y == rep x = rep y
x + y == per merge(#2 < #1, rep x, rep y)$Rep
n:NNI * x:% ==
zero? n => 0
x + (subtractIfCan(n,1) :: NNI) * x
remv(i: PI,x: %): Maybe % ==
member?(i,rep x) => just per remove(i, rep x)$Rep
nothing
subtractIfCan(x, y) ==
zero? x =>
zero? y => just 0
nothing
zero? y => just x
aa := remv(first rep y,x)
aa case nothing => nothing
subtractIfCan(aa@%, per rest rep y)
powers x ==
l := rep x
r: List Pair(PI,PI) := nil
while not empty? l repeat
i := first l
-- Now, count how many times the item `i' appears in `l'.
-- Since parts of partitions are stored in decreasing
-- order, we only need to scan the rest of the list until
-- we hit a different number.
n: PI := 1
while not empty?(l := rest l) and i = first l repeat
n := n + 1
r := cons(pair(i,n), r)
reverse! r
conjugate x == per conjugate(rep x)$PartitionsAndPermutations
mkterm(i1: I,i2: I): OUT ==
i2 = 1 => (i1 :: OUT) ** (" " :: OUT)
(i1 :: OUT) ** (i2 :: OUT)
mkexp1(lli: L Pair(PI,PI)): L OUT ==
empty? lli => nil
li := first lli
empty?(rest lli) and second(li) = 1 =>
[first(li) :: OUT]
cons(mkterm(first li,second li),mkexp1(rest lli))
coerce(x:%):OUT ==
empty? rep x => rep(x)::OUT
paren(reduce("*",mkexp1(powers x)))
pdct x ==
*/[factorial(second a) * (first(a) ** second(a))
for a in powers x] :: PI
@
\section{domain SYMPOLY SymmetricPolynomial}
<<domain SYMPOLY SymmetricPolynomial>>=
)abbrev domain SYMPOLY SymmetricPolynomial
++ Description:
++ This domain implements symmetric polynomial
SymmetricPolynomial(R:Ring) == PolynomialRing(R,Partition) add
Term == Record(k:Partition,c:R)
Rep:= List Term
-- override PR implementation because coeff. arithmetic too expensive (??)
if R has EntireRing then
(p1:%) * (p2:%) ==
null p1 => 0
null p2 => 0
zero?(p1.first.k) => p1.first.c * p2
one? p2 => p1
+/[[[t1.k+t2.k,t1.c*t2.c]$Term for t2 in p2]
for t1 in reverse(p1)]
-- This 'reverse' is an efficiency improvement:
-- reduces both time and space [Abbott/Bradford/Davenport]
else
(p1:%) * (p2:%) ==
null p1 => 0
null p2 => 0
zero?(p1.first.k) => p1.first.c * p2
one? p2 => p1
+/[[[t1.k+t2.k,r]$Term for t2 in p2 | (r:=t1.c*t2.c) ~= 0]
for t1 in reverse(p1)]
-- This 'reverse' is an efficiency improvement:
-- reduces both time and space [Abbott/Bradford/Davenport]
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
<<domain PRTITION Partition>>
<<domain SYMPOLY SymmetricPolynomial>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
