

last edited 15 years ago 
1  
Editor:
Time: 2007/11/17 22:22:50 GMT8 

Note: 
changed:  It would be nice if 'PERMGRP' could generate the cycle indicator polynomial for a given permutation group. Furthermore, it should be able to list all the elements of a given permutation group as a stream. Finally, it should be able to reduce the set of given generators to a minimal set. For the first two items I have a brute force solution, there might be better ones, of course. I guess that all of this will be merged into the combinat package. Here is the code, for those who want to play with it::  generate all permutations given the generators  multiply each of the elements in result by each of the generators listaux(gens: List PERM INT, result: List PERM INT): List PERM INT == removeDuplicates(append(result, reduce(append, [[p*q for q in result] for p in gens]))) list(gens: List PERM INT): List PERM INT == result := [1$PERM INT] lold := 0 lnew := 1 while lnew > lold repeat result := listaux(gens, result) lold := lnew lnew := #(result)$List PERM INT result  collect cyclePartitions into the cycle indicator polynomial cyclePoly(lp: List Partition, vars: List POLY INT): POLY INT == n := #vars p := lp.1 c := 1 result := 0 for q in rest lp repeat if q = p then c := c+1 else fix := nreduce(+, p::List INT, 0) result := result + c * reduce((a,b) +> a*b, _ [vars.i for i in p::List INT], _ (vars.1)**fix) p := q c := 1 fix := nreduce(+, p::List INT) result := result + c * reduce((a,b) +> a*b, _ [vars.i for i in p::List INT], _ (vars.1)**fix) Martin
It would be nice if PERMGRP
could generate the cycle indicator polynomial for a given permutation group.
Furthermore, it should be able to list all the elements of a given permutation group as a stream.
Finally, it should be able to reduce the set of given generators to a minimal set.
For the first two items I have a brute force solution, there might be better ones, of course. I guess that all of this will be merged into the combinat package. Here is the code, for those who want to play with it:
 generate all permutations given the generators  multiply each of the elements in result by each of the generators listaux(gens: List PERM INT, result: List PERM INT): List PERM INT == removeDuplicates(append(result, reduce(append, [[p*q for q in result] for p in gens]))) list(gens: List PERM INT): List PERM INT == result := [1$PERM INT] lold := 0 lnew := 1 while lnew > lold repeat result := listaux(gens, result) lold := lnew lnew := #(result)$List PERM INT result  collect cyclePartitions into the cycle indicator polynomial cyclePoly(lp: List Partition, vars: List POLY INT): POLY INT == n := #vars p := lp.1 c := 1 result := 0 for q in rest lp repeat if q = p then c := c+1 else fix := nreduce(+, p::List INT, 0) result := result + c * reduce((a,b) +> a*b, _ [vars.i for i in p::List INT], _ (vars.1)**fix) p := q c := 1 fix := nreduce(+, p::List INT) result := result + c * reduce((a,b) +> a*b, _ [vars.i for i in p::List INT], _ (vars.1)**fix)
Martin