1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
|
\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra prtition.spad}
\author{William H. Burge}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain PRTITION Partition}
<<domain PRTITION Partition>>=
)abbrev domain PRTITION Partition
++ Domain for partitions of positive integers
++ Author: William H. Burge
++ Date Created: 29 October 1987
++ Date Last Updated: 23 Sept 1991
++ Keywords:
++ Examples:
++ References:
Partition: Exports == Implementation where
++ 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}.
L ==> List
I ==> Integer
OUT ==> OutputForm
NNI ==> NonNegativeInteger
UN ==> Union(%,"failed")
Exports ==> Join(OrderedCancellationAbelianMonoid,
ConvertibleTo List Integer) with
partition: L I -> %
++ partition(li) converts a list of integers li to a partition
powers: L I -> L L I
++ powers(li) returns a list of 2-element lists. For each 2-element
++ list, the first element is an entry of li and the second
++ element is the multiplicity with which the first element
++ occurs in li. There is a 2-element list for each value
++ occurring in l.
pdct: % -> I
++ \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
coerce:% -> List Integer
++ coerce(p) coerces a partition into a list of integers
Implementation ==> add
import PartitionsAndPermutations
Rep := List Integer
0 == nil()
coerce (s:%) == s pretend List Integer
convert x == copy(x pretend L I)
partition list == sort(#2 < #1,list)
x < y ==
empty? x => not empty? y
empty? y => false
first x = first y => rest x < rest y
first x < first y
x = y ==
EQUAL(x,y)$Lisp
-- empty? x => empty? y
-- empty? y => false
-- first x = first y => rest x = rest y
-- false
x + y ==
empty? x => y
empty? y => x
first x > first y => concat(first x,rest(x) + y)
concat(first y,x + rest(y))
n:NNI * x:% == (zero? n => 0; x + (subtractIfCan(n,1) :: NNI) * x)
dp: (I,%) -> %
dp(i,x) ==
empty? x => 0
first x = i => rest x
concat(first x,dp(i,rest x))
remv: (I,%) -> UN
remv(i,x) == (member?(i,x) => dp(i,x); "failed")
subtractIfCan(x, y) ==
empty? x =>
empty? y => 0
"failed"
empty? y => x
(aa := remv(first y,x)) case "failed" => "failed"
subtractIfCan((aa :: %), rest y)
li1 : L I --!! 'bite' won't compile without this
bite: (I,L I) -> L I
bite(i,li) ==
empty? li => concat(0,nil())
first li = i =>
li1 := bite(i,rest li)
concat(first(li1) + 1,rest li1)
concat(0,li)
li : L I --!! 'powers' won't compile without this
powers l ==
empty? l => nil()
li := bite(first l,rest l)
concat([first l,first(li) + 1],powers(rest li))
conjugate x == conjugate(x pretend Rep)$PartitionsAndPermutations
mkterm: (I,I) -> OUT
mkterm(i1,i2) ==
i2 = 1 => (i1 :: OUT) ** (" " :: OUT)
(i1 :: OUT) ** (i2 :: OUT)
mkexp1: L L I -> L OUT
mkexp1 lli ==
empty? lli => nil()
li := first lli
empty?(rest lli) and second(li) = 1 =>
concat(first(li) :: OUT,nil())
concat(mkterm(first li,second li),mkexp1(rest lli))
coerce(x:%):OUT ==
empty? (x pretend Rep) => coerce(x pretend Rep)$Rep
paren(reduce("*",mkexp1(powers(x pretend Rep))))
pdct x ==
*/[factorial(second a) * (first(a) ** (second(a) pretend NNI))
for a in powers(x pretend Rep)]
@
\section{domain SYMPOLY SymmetricPolynomial}
<<domain SYMPOLY SymmetricPolynomial>>=
)abbrev domain SYMPOLY SymmetricPolynomial
++ Description:
++ This domain implements symmetric polynomial
SymmetricPolynomial(R:Ring) == PolynomialRing(R,Partition) add
Term:= Record(k:Partition,c:R)
Rep:= List Term
-- override PR implementation because coeff. arithmetic too expensive (??)
if R has EntireRing then
(p1:%) * (p2:%) ==
null p1 => 0
null p2 => 0
zero?(p1.first.k) => p1.first.c * p2
-- one? p2 => p1
(p2 = 1) => p1
+/[[[t1.k+t2.k,t1.c*t2.c]$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
(p2 = 1) => 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}
|
