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\documentclass{article}
\usepackage{axiom}
\begin{document}
\title{\$SPAD/src/algebra tableau.spad}
\author{William H. Burge}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain TABLEAU Tableau}
<<domain TABLEAU Tableau>>=
)abbrev domain TABLEAU Tableau
++ Author: William H. Burge
++ Date Created: 1987
++ Date Last Updated: 23 Sept 1991
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: Young tableau
++ References:
++ Description:
++ The tableau domain is for printing Young tableaux, and
++ coercions to and from List List S where S is a set.
Tableau(S:SetCategory):Exports == Implementation where
++ The tableau domain is for printing Young tableaux, and
++ coercions to and from List List S where S is a set.
L ==> List
I ==> Integer
NNI ==> NonNegativeInteger
OUT ==> OutputForm
V ==> Vector
fm==>formMatrix$PrintableForm()
Exports ==> with
tableau : L L S -> %
++ tableau(ll) converts a list of lists ll to a tableau.
listOfLists : % -> L L S
++ listOfLists t converts a tableau t to a list of lists.
coerce : % -> OUT
++ coerce(t) converts a tableau t to an output form.
Implementation ==> add
Rep := L L S
tableau(lls:(L L S)) == lls pretend %
listOfLists(x:%):(L L S) == x pretend (L L S)
makeupv : (NNI,L S) -> L OUT
makeupv(n,ls)==
v:=new(n,message " ")$(List OUT)
for i in 1..#ls for s in ls repeat v.i:=box(s::OUT)
v
maketab : L L S -> OUT
maketab lls ==
ll : L OUT :=
empty? lls => [[empty()]]
sz:NNI:=# first lls
[blankSeparate makeupv(sz,i) for i in lls]
pile ll
coerce(x:%):OUT == maketab listOfLists x
@
\section{package TABLBUMP TableauxBumpers}
<<package TABLBUMP TableauxBumpers>>=
)abbrev package TABLBUMP TableauxBumpers
++ Author: William H. Burge
++ Date Created: 1987
++ Date Last Updated: 23 Sept 1991
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords: Young tableau
++ References:
++ Description:
++ TableauBumpers implements the Schenstead-Knuth
++ correspondence between sequences and pairs of Young tableaux.
++ The 2 Young tableaux are represented as a single tableau with
++ pairs as components.
TableauxBumpers(S:OrderedSet):T==C where
L==>List
ST==>Stream
B==>Boolean
ROW==>Record(fs:B,sd:L S,td:L L S)
RC==>Record(f1:L S,f2:L L L S,f3:L L S,f4:L L L S)
PAIR==>L S
T== with
bumprow:((S,S)->B,PAIR,L PAIR)->ROW
++ bumprow(cf,pr,r) is an auxiliary function which
++ bumps a row r with a pair pr
++ using comparison function cf, and returns a record
bumptab:((S,S)->B,PAIR,L L PAIR)->L L PAIR
++ bumptab(cf,pr,t) bumps a tableau t with a pair pr
++ using comparison function cf, returning a new tableau
bumptab1:(PAIR,L L PAIR)->L L PAIR
++ bumptab1(pr,t) bumps a tableau t with a pair pr
++ using comparison function \spadfun{<},
++ returning a new tableau
untab: (L PAIR,L L PAIR)->L PAIR
++ untab(lp,llp) is an auxiliary function
++ which unbumps a tableau llp,
++ using lp to accumulate pairs
bat1:L L PAIR->L PAIR
++ bat1(llp) unbumps a tableau llp.
++ Operation bat1 is the inverse of tab1.
bat:Tableau(L S)->L L S
++ bat(ls) unbumps a tableau ls
tab1:L PAIR->L L PAIR
++ tab1(lp) creates a tableau from a list of pairs lp
tab:L S->Tableau(L S)
++ tab(ls) creates a tableau from ls by first creating
++ a list of pairs using \spadfunFrom{slex}{TableauBumpers},
++ then creating a tableau using \spadfunFrom{tab1}{TableauBumpers}.
lex:L PAIR->L PAIR
++ lex(ls) sorts a list of pairs to lexicographic order
slex:L S->L PAIR
++ slex(ls) sorts the argument sequence ls, then
++ zips (see \spadfunFrom{map}{ListFunctions3}) the
++ original argument sequence with the sorted result to
++ a list of pairs
inverse:L S->L S
++ inverse(ls) forms the inverse of a sequence ls
maxrow:(PAIR,L L PAIR,L PAIR,L L PAIR,L L PAIR,L L PAIR)->RC
++ maxrow(a,b,c,d,e) is an auxiliary function for mr
mr:L L PAIR->RC
++ mr(t) is an auxiliary function which
++ finds the position of the maximum element of a tableau t
++ which is in the lowest row, producing a record of results
C== add
cf:(S,S)->B
bumprow(cf,x:(PAIR),lls:(L PAIR))==
if null lls
then [false,x,[x]]$ROW
else (y:(PAIR):=first lls;
if cf(x.2,y.2)
then [true,[x.1,y.2],cons([y.1,x.2],rest lls)]$ROW
else (rw:ROW:=bumprow(cf,x,rest lls);
[rw.fs,rw.sd,cons(first lls,rw.td)]$ROW ))
bumptab(cf,x:(PAIR),llls:(L L PAIR))==
if null llls
then [[x]]
else (rw:ROW:= bumprow(cf,x,first llls);
if rw.fs
then cons(rw.td, bumptab(cf,rw.sd,rest llls))
else cons(rw.td,rest llls))
bumptab1(x,llls)==bumptab(#1<#2,x,llls)
rd==> reduce$StreamFunctions2(PAIR,L L PAIR)
tab1(lls:(L PAIR))== rd([],bumptab1,lls::(ST PAIR))
srt==>sort$(PAIR)
lexorder:(PAIR,PAIR)->B
lexorder(p1,p2)==if p1.1=p2.1 then p1.2<p2.2 else p1.1<p2.1
lex lp==(sort$(L PAIR))(lexorder(#1,#2),lp)
slex ls==lex([[i,j] for i in srt(#1<#2,ls) for j in ls])
inverse ls==[lss.2 for lss in
lex([[j,i] for i in srt(#1<#2,ls) for j in ls])]
tab(ls:(PAIR))==(tableau tab1 slex ls )
maxrow(n,a,b,c,d,llls)==
if null llls or null(first llls)
then [n,a,b,c]$RC
else (fst:=first first llls;rst:=rest first llls;
if fst.1>n.1
then maxrow(fst,d,rst,rest llls,cons(first llls,d),rest llls)
else maxrow(n,a,b,c,cons(first llls,d),rest llls))
mr llls==maxrow(first first llls,[],rest first llls,rest llls,
[],llls)
untab(lp, llls)==
if null llls
then lp
else (rc:RC:=mr llls;
rv:=reverse (bumptab(#2<#1,rc.f1,rc.f2));
untab(cons(first first rv,lp)
,append(rest rv,
if null rc.f3
then []
else cons(rc.f3,rc.f4))))
bat1 llls==untab([],[reverse lls for lls in llls])
bat tb==bat1(listOfLists tb)
@
\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 TABLEAU Tableau>>
<<package TABLBUMP TableauxBumpers>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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