Dear Ralf, Peter,
Peter: still no way to get aldor "extend" working for axiom, I guess?
last week I tried, for the first time, to use aldor-combinat to generate some
objects, namely partitions of type B and D. Partitions of type A are the usual
set partitions you know. A partition of type B is a set partition of
{+1,-1,+2, -2,...,+n, -n}
such that
1) for every block, the negation of the block is also a block. I.e., if
{1,2,-3} is a block, then {-1,-2,3} must also be a block
2) there is at most one block, called the zero block, which contains both i and
-i for some i. It then follows from (1) that it is a set of the form
{+a, -a, +b, -b,...}
Below is naive code to generate this objects using axiom and aldor-combinat.
The reason I send you this, is to give you a feeling how I might want to use
our code, and that it is not quite optimal yet, due to the incredible number of
coercions necessary. To make things easier, I added a coercion from SetSpecies
to Set, lest I go crazy.
It also proves that you are right in saying that
coerce: % -> Record(subset: SetSpecies L ,rest: SetSpecies L)
in Subset is not optimal.
Of course, most of the awkwardness would go away if we had "extend" usable for
Axiom, but that seems still out of reach.
I do not want to say that what we have is no good. Only, to make it
productive, there is some (probably boring) work to do. One thing is that
since so much is based on SetSpecies, we should really make it work well.
I'm looking forward to see you and Christian next Monday,
Martin
--)cd ~/combinat/branches/iso-experiment/combinat/src
--)re ../lib/combinat.input
--)cd ~/martin/TeXSource/Mathematik/jakob
-- produce all type B partitions
REC ==> Record(part: Partition ACINT, zero: SetSpecies ACINT, negs: SetSpecies
ACINT)
partitionsB n ==
lab: SetSpecies ACINT := set [i::ACINT for i in 1..n]
all := [structures(lab)$Partition ACINT]$ACList Partition ACINT::List
Partition ACINT
res: List REC := []
for p in all repeat
llp := p::SetSpecies SetSpecies ACINT::ACList SetSpecies ACINT::List
SetSpecies ACINT
-- add an empty block, for the case where there is no zero block
llp := cons(set([]::List ACINT)$SetSpecies ACINT, llp)
-- select a zero block
for z in llp repeat
-- all numbers but the very first in each block may be barred
pos := z
for b in rest llp repeat
pos := cons(first b, pos)
r: SetSpecies ACINT := set (parts difference(lab::Set ACINT,
pos::Set ACINT))
subs := [structures(r)$Subset ACINT]$ACList Subset ACINT::List
Subset ACINT
for s in subs repeat
t := s::Record(subset: SetSpecies ACINT ,rest: SetSpecies ACINT)
res := cons([p, z, t.subset]::REC, res)
res
-- [# partitionsB n for n in 0..7] produces Dowling numbers
-- http://www.research.att.com/~njas/sequences/A007405
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