#
# extract_jacks - extract Jack functions from a remember table in SF
# restore_jacks - install a list of Jack functions into a remember table
#
# Calling sequence:
#  extract_jacks(n)
#  restore_jacks(n,Jlist)
#
# Parameters:
#    n   - a positive integer
#  Jlist - a list of Jack functions of degree n (as given by extract_jacks)
#
# Assume that the Jack symmetric function basis has been defined during an
# SF session, using 'J' as the basis name and 'a' as the Jack parameter.
# In addition, suppose that the e-function expansions for all Jack functions
# J[mu] of degree n have been computed--either directly, or using the
# accelerated code in SF/examples/jacks.
#
# Given the above conditions, extract_jacks(n) will extract a list (in a
# highly compressed format) of the e-function expansions of Jack functions
# of degree n from their remember table. The resulting list can then be
# saved to a file.
#
# Given that Jlist is the list produced by extract_jacks(n), then
# restore_jacks(n,Jlist) will reverse the procedure, storing the data from
# the list into the remember table (and if necessary, using add_basis to
# define the Jack function basis in the current SF session).
#
# Examples:
#   with(SF);
# Define the Jack functions:
#   add_basis(J,mu->zee(mu,a),mu->hooks(mu,a));
# This builds the remember table for degree 4:
#   for mu in Par(4) do toe(J[op(mu)]) od;
# Now extract the table in compressed form:
#   Jack4:=extract_jacks(4);
# For faster building of the table, use SF/examples/jacks:
#   read cat(HomeLib,`/SF/examples/jacks`);   #assumes the Unix Edition
#   jacks(9,verbose);
#   Jack9:=extract_jacks(9):
# Now save the results to a file:
#   save Jack4,Jack9,`filename.m`;
#
# To restore the data in a new Maple session:
#   with(SF);
#   read `filename.m`; # remember also to read in 'jacktools'
#   restore_jacks(4,Jack4);
#   restore_jacks(9,Jack9);
#   toe(J[4,3,2]); 

extract_jacks:=proc(n) local sp,N,i,monid,vars;
  sp:=SF['Par'](n); N:=nops(sp);
  monid:=table([seq(convert(map(x->cat('e',x),sp[i]),`*`)=i,i=1..N)]);
  vars:=[seq(cat('e',i),i=1..n)];
  monid:=[seq(`jacks/compr`(sp[i],i,vars,monid),i=1..N)];
end;

restore_jacks:=proc(n,Jlist) local par2e,sp,i,f;
  if not member('J[]',`SF/Bases`) then
    SF['add_basis'](J,mu->SF['zee'](mu,a),mu->SF['hooks'](mu,a))
  fi;
  `SF/added`(J,[]);  # prevent clobbering of the remember table
  sp:=SF['Par'](n);
  par2e:=map(convert,subs({seq(i=cat('e',i),i=1..n)},sp),`*`);
  for i to nops(sp) do
    f:=convert(zip((x,y)->`jacks/unwrap`(x)*y,Jlist[i],par2e),`+`);
    `SF/added/gs`(J,sp[i]):=[0,f]
  od; NULL
end;

# compress a single J[mu]

`jacks/compr`:=proc(mu,i,vars,Mid) local v,co,tm;
  co:=[coeffs(`SF/added`(J,mu),vars,'tm')];
  tm:=[seq(Mid[v],v=[tm])];
  co:=map(`jacks/compr/poly`,co);
  subsop(op(zip((x,y)->x=y,tm,co)),[0$i]);
end:

# compress a (nonzero) univariate polynomial

`jacks/compr/poly`:=proc(f) local c,k,g,i;
  g:=expand(f); c:=igcd(coeffs(g,a));
  k:=ldegree(g,a); g:=g/c;
  [k,c,seq(coeff(g,a,i),i=k..degree(g,a))];
end:

`jacks/unwrap`:=proc(v) local j;
  if v=0 then 0 else
    v[2]*convert([seq(v[j]*a^(j+v[1]-3),j=3..nops(v))],`+`) fi
end: