#Homework by Mike Murr
#OK to post
#hw14

#BZ(L,j): Implements “Bijecting Euler’s Partition Recurrence” by
#         David Bressoud and Doron Zeilberger, January, 1985.
#         Inputs a partition L and an integer j.
#         Outputs a partition Lprime and an integer jprime,
#         where jprime has opposite parity from j and 
#         Lprime and jprime satisfy an additional property.

BZ:=proc(L,j) local a, t, i, n1, lhs, rhs, Lprime, jprime;

if nops(L) = 0 then
  print(`L is empty`);
  return(Fail);
fi;

a:=(3*j^2+j)/2;

t:=nops(L);

n1:=sum(L[i],i=1..t);

lhs:=t + 3*j;
rhs:=L[1];

print(a);
print(n1);

if lhs>=rhs then
#  Lprime[1]:= t + 3*j - 1;
#  for i from 1 to t do
#    Lprime[i+1] := L[i] - 1;  
#  od;
  Lprime := [t + 3*j - 1, seq(L[i]-1,i=1..t)];
  jprime := j - 1;
fi;

if lhs < rhs then
  Lprime:= [seq(L[i]+1,i=2..t), seq(1,i=1..L[1]-3*j-t-1)];
  jprime:= j + 1;
fi;

return(Lprime,jprime);

end proc;

BZ([5,5,4,3,2],1);

BZ([7,4,4,3,2,1],0);

#Glaisher(L): Implements Glaisher’s bijection from a partition with odd parts to a
#             partition with distinct parts.
#             One reference for this is “Partition Bijections, A Survey,” by Igor Pak.
#             math.ucla.edu/~pak/papers/psurvey.pdf
#             Glaisher’s is on page 20 of the survey.
Glaisher:=proc(L) local L0,count, odds, multOdds, i, index, j, mult, r, output;

L0:=[op(L),0];

if nops(L) = 0 then
  return([]);
fi;

if nops(L) = 1 then
  return(L);
fi;

count:=1;
index:=1;
odds:=[];
multOdds:=[];
output:=[];
for i from 1 to nops(L0)-1 do
  if L0[i+1]=L0[i] then
    count:=count+1;
#    print(i);
  else
    odds:=[op(odds), L[i]];
    multOdds:=[op(multOdds), count];
    count:=1;
    index:=index+1;
  fi;
od;
#print(odds);
#print(multOdds);

# odds:= each odd part
# multOdds:= the multiplicity of each odd part

for i from 1 to nops(odds) do

  mult:=multOdds[i];
  r:=0;
  while mult > 0 do
    if mult mod 2 = 1 then
      output:=[op(output), odds[i]*2^r];
    fi;
    mult:= iquo(mult,2);
    r:=r+1;
  od;
od;
output:=sort(output,`>`);
print(output);
  

end proc;

Glaisher([5,5,5,3,3]);

Glaisher([7,7,7,7,1,1,1,1]);