# HW 24
# Jike Liu

read C24.txt;

# Question 1:
Sc2:=proc(x,L) local n,L1,i,j; n:=convert(L,`+`): L1:=[op(L),0$(n-nops(L))]: expand(det(matrix([seq([seq(hkn(L1[i]-i+j,n,x),j=1..n)],i=1..n)]))): end:


# Question 2: 

Conj:=proc(L) local k,C1,i1,L1,i:
option remember:
if L=[] then
RETURN([]):
fi:
k:=nops(L):
L1:=[seq(L[i]-1,i=1..k)]:

for i1 from 1 to nops(L1) while L1[i1]>0 do od:
i1:=i1-1:

L1:=[op(1..i1,L1)]:

C1:=Conj(L1):
[k,op(C1)]:
end:

Sc3:=proc(x,L) local n,C,i,j; n:=convert(L,`+`): C:=[op(Conj(L)),0$(n-nops(Conj(L)))]: expand(det(matrix([seq([seq(ekn(C[i]-i+j,n,x),j=1..n)],i=1..n)]))): end:


# Question 3:

# ------------------------------------------------------------
# Comparative shopping for Sc, Sc1, Sc2, Sc3 at n=5
# ------------------------------------------------------------

P:=Par(5):

L0:=[seq(expand(Sc(x,P[i])),  i=1..nops(P))]:
L1:=[seq(expand(Sc1(x,P[i])), i=1..nops(P))]:
L2:=[seq(expand(Sc2(x,P[i])), i=1..nops(P))]:
L3:=[seq(expand(Sc3(x,P[i])), i=1..nops(P))]:

evalb(L0=L1), evalb(L0=L2), evalb(L0=L3);


# ------------------------------------------------------------
# Timing each method at n=5
# ------------------------------------------------------------

t0:=time():
P:=Par(5):
A0:=[seq(expand(Sc(x,P[i])),i=1..nops(P))]:
time()-t0;

t1:=time():
P:=Par(5):
A1:=[seq(expand(Sc1(x,P[i])),i=1..nops(P))]:
time()-t1;

t2:=time():
P:=Par(5):
A2:=[seq(expand(Sc2(x,P[i])),i=1..nops(P))]:
time()-t2;

t3:=time():
P:=Par(5):
A3:=[seq(expand(Sc3(x,P[i])),i=1..nops(P))]:
time()-t3;

# Output:
#                       true, true, true
#
                             0.016

#                            0.017

#                            0.001

#                            0.001



# Question 4: 

SSYT:=proc(L,a) local k,MR,R,r,L1,a1,S,T,T1,i,j;
option remember:

if a<nops(L) then
 RETURN({}):
fi:

if L=[] then
 RETURN({}):
fi:

k:=nops(L):

if k=1 then
 if L[1]=1 then
   RETURN({[[a]]}):
 else
   RETURN({seq(seq([[op(T[1]),a]], T in SSYT([L[1]-1],a1)), a1=1..a)}):
 fi:
fi:

MR:=MaxRims(L):
R:=SubSeqs(MR) minus {[0$k]}:
S:={}:

for r in R do
 L1:=[seq(L[i]-r[i],i=1..k)]:
 L1:=KickZero(L1):

 for a1 from nops(L1) to a-1 do
  for T in SSYT(L1,a1) do
   T1:=[op(T), seq([],j=1..k-nops(T))]:
   S:=S union {[seq([op(T1[i]),a$r[i]],i=1..k)]}:
  od:
 od:
od:

S:
end: